(     B  ERKELEY 

I    LuiRARY 


MATH/:5TAT, 


MATH/STAT» 


^ 


GIROLAMO  SACCHERFS 

EUCLIDES  VINDICATUS 


EDITED  AND  TRANSLATED  BY 

GEORGE  BRUCE  HALSTED 

A.M.,  PRINCETON,  PH.D.,  JOHNS  HOPKINS 


CHICAGO  LONDON 

THE  OPEN  COURT  PUBLISHING  COMPANY 

1920 


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MArH 


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copyright  by 

The  Open  Court  Publishing  Company 

1920 


TABLE  OF  CONTENTS. 

PAGK 

Translator's  Preface  vii 

Introduction  xv 

Saccheri's  Title-Page    3 

Preface  to  the  Reader  5 

In  Place  of  an  Index  11 

PART  I. 

Proposition   1 19 

Proposition  II 21 

Proposition  III 21 

Corollary  1 25 

Corollary  II 27 

Corollary  III 27 

Proposition  IV 27 

Definitions 29 

Proposition  V 29 

Proposition  VI 33 

Proposition  VII 37 

Proposition  VIII 39 

Proposition  IX 41 

Proposition  X 43 

Proposition  XI 45 

Proposition  XII 49 

Proposition  XIII 53 

Schohon  1 55 

SchoHon  II 55 

Proposition  XIV 59 

Proposition  XV 61 

Corollary 65 

Proposition  XVI 67 

Corollary  69 

Proposition  XVII 69 

Schohon  1 73 

SchoHon  II 73 

iii 


O.SMV 


rAOB 

Proposition  XVIII 75 

Proposition  XIX 75 

Proposition  XX 79 

Proposition  XXI 81 

Corollary 81 

Scholion  1 83 

Scholion  II 87 

Scholion  III 101 

Scholion  IV 109 

Proposition  XXII 113 

Proposition  XXIII 117 

Corollary  1 119 

Scholion  121 

Corollary  II 121 

Proposition  XXIV 125 

Corollary  127 

Scholion 129 

Proposition  XXV 131 

Corollary  I 137 

Corollary  II 137 

Proposition  XXVI 139 

Corollary  1 141 

Corollary  II 141 

Corollary  III 143 

Proposition  XXVII 143 

Scholion  I , 147 

Scholion  II 149 

Scholion  III 149 

Proposition  XXVIII 149 

Corollary 157 

Proposition  XXIX 157 

Corollaryl 159 

Corollary  II 159 

Proposition  XXX 161 

Corollary 165 

Proposition  XXXI 167 

Proposition  XXXII 169 

Proposition  XXXIII 173 

Lemma  1 173 

Corollaryl 177 

Corollary  II 179 

Lemma  II 179 

Corollary 195 

Lemma  III 197 

LemmalV 199 

iv 


rAos 

Scholion 201 

Lemma  V 201 

Corollary  . 205 

Scholion 207 

PART  II. 

Proposition  XXXIV  209 

Proposition  XXXV 211 

Proposition  XXXVI 213 

Corollary  215 

Proposition  XXXVII 215 

Scholion  1 221 

Scholion  II 225 

Proposition  XXXVIII 225 

Scholion 229 

Proposition  XXXIX 231 

Scholion 233 

Notes 243 

Translator's  Subject  Index  245 

Translator's  Index  of  Proper  Names 246 


TRANSLATOR'S  PREFACE. 

Saccheri  was  discovered  accidentally  by  Father  Man- 
ganotti,  S.  J.,  in  1889,  a  discovery  which  could  not  have 
happened  on  the  Westem  Continent,  as  it  was  only  my  in- 
stant  reahzation  of  the  treasure  found  which  then,  through 
my  friend  Prof.  Paul  Mansion,  drew  to  me  overseas  the 
first  Saccheri  ever  to  have  crossed  the  ocean.  This,  upon 
my  securing  Earl  Stanhope's  copy,  I  sent  back,  to  be  whelmed 
in  a  common  death  with  the  dearly  beloved  Mansion  and 
the  magnificent  Alberto  Pascal. 

How  rare,  practically  extinct,  Saccheri  has  been  is  illus- 
trated  by  the  fact  that  the  greatest  of  living  non-EucHdeans, 
Prof.  Paul  Barbarin,  though  resident  in  the  world's  capital, 
Paris,  the  modem  Alexandria,  yet  sent,  as  I  had  done  before 
him,  to  Mansion  f  or  a  gHmpse  of  this  pearl  of  great  price ; 
and  again  by  the  fact  that  no  one  has  remarked  that  the 
erudite  and  accurate  Sir  Thomas  Heath,  in  his  three-volumed 
masterpiece,  has  Hsted  Saccheri's  Euclides  znndicatus  as  a 
Latin  edition  of  EucHd,  caHing  Saccheri  its  editor,  a  mis- 
take  so  gross  it  could  never  have  been  made  by  any  one 
who  had  been  privileged  to  see  Saccheri's  diadem. 

Does  it  not  seem  to  be  the  irony  of  fate  that  the  only 
existing  copy  of  the  posthumous  edition  of  Saccheri's  won- 
deriul  Logica  demonstrativa  reposes  in  the''StadtbibHothek" 
of  the  foreign  city  built  by  Nero's  mother — Colonia  Agrip- 
pina  on  the  banks  of  the  Rhine?  May  the  present  volume 
help  to  avert  from  it  the  imminent  peril  of  perishing  for- 


ever,  if  indeed  the  waves  of  oblivion  have  not  already  closed 
over  it. 

Manganotti  planned  a  reproduction  of  the  Euclides 
vindicatus;  Alberto  Pascal  planned  a  complete  Italian  ver- 
sion,  saying,  "We  cannot  call  an  ItaHan  translation  that  pub- 
lished  by  G.  Boccardini,  which  with  its  curious  abbreviated 
demonstrations,  with  its  suppression  of  whole  pages  and 
with  its  prudent  transcription  entire  of  Latin  passages  not 
easily  translatable,  truly  knows  not  itself  what  it  is." 

Death  vetoed  these  plans. 

Meanwhile  legends  grow  up  and  persist  about  Saccheri 
himself .  For  example,  no  less  an  authority  than  a  president 
of  the  New  York  Mathematical  Society,  misled  by  scraps 
of  Saccheri's  Latin  given  by  Behrami,  has  in  an  article  "On 
the  Early  History  of  Non-Euclidean  Geometry"  {Bull.  N. 
Y.  Math.  Soc,  Vol.  II,  pp.  144-147)  the  sentence:  "He  con- 
f  essed  to  a  distracting  heretical  tendency  on  his  part  in  f  avor 
of  the  hypothesis  anguli  acuti,  a  tendency  against  which, 
however,  he  kept  up  a  perpetual  struggle  (diuturnum  proe- 
lium)."  And  this  sentence  is  quoted  without  dissent  in 
The  Monist,  Vol.  IV,  p.  489.  Contrast  the  truth  of  the 
matter  by  glancing  at  Saccheri's  ^lndicis  loco"  given  below, 
where  the  two  quoted  words  occur  (last  sentence  on  p.  xii). 

But  inaccessible  as  he  was,  and  legendary  as  he  became, 
Saccheri's  immortahty  was  already  assured.  The  name  of 
his  friend  Ceva  is  carried  by  Ceva's  Theorem. 

Coolidge  in  his  N on-Euclidean  Geometry  names  Sac- 
cheri's  the  theorem:  "In  an  isosceles  birectangular  quadri- 
lateral  a  line  through  the  middle  point  of  the  side  adjacent 
to  both  right  angles,  which  is  perpendicular  to  the  line  of 
that  side,  will  be  perpendicular  to  the  line  of  the  opposite 
side  and  pass  through  its  middle  point.  The  other  two 
angles  of  the  quadrilateral  are  mutualfy  congruent."  In 
1905  Bonola  calls  Saccheri's  the  theorem :  "If  the  angle-sum 
in  one  triangle  be  equal  to,  greater  than,  or  less  than  two 
right  angles,  so  will  it  be  in  every  triangle."  But  in  1896 
Mansion  had  admirably  chosen  as  Theorkme  de  Saccheri: 
"Dans  l'hypothese  ou  la  somme  des  angles  d'un  triangle  est 


inferieure  a  deux  droits,  deux  droites  d'un  plan  se  ren- 
contrent,  ou  sont  asymptotes  Tune  de  Tautre,  ou  ont  une 
perpendiculaire  commune  a  partir  de  laquelle  elles  di- 
vergent." 

As  substitute  for  Euclid's,  Borelli  in  1658  proposed  the 
definition:  "Parallels  are  coplanar  straights  with  a  common 
perpendicular."  In  1756  the  "famous"  Robert  Simson  gave, 
as  basis  for  a  "proof "  of  Euclid's  Postulate,  a  new  Axiom : 
"A  straight  line  cannot  approach  toward,  and  then  recede 
from,  a  straight  line  without  cutting  it."  But  in  Saccheri's 
Theorem  not  only  occur,  for  the  first  time  in  the  world, 
straight  lines  which  are  asymptotes  one  of  another,  but 
two  perpendiculars  to  a  straight  line  spread  azvay  from 
each  other  as  they  go  out ;  their  points  at  two  inches  f rom 
the  straight  line  are  farther  apart  than  their  points  one  inch 
from  the  line. 

Again,  Geminos  (circa  100  B.  C.)  defined  parallels  as 
straights  everywhere  equidistant,  but  Giordano  Vitale  da 
Bitonto  (1680)  saw  that  this  presupposed  the  assumption 
of  Clavius,  1574,  that  a  line  coplanar  with  a  straight  and 
everywhere  equidistant  from  it  is  itself  straight;  so  using 
a  figure  found  in  Clavius,  made  by  joining  together  the  ends 
of  two  equal  perpendiculars  upon  a  straight,  he  tried  to 
prove  this  join  everywhere  equidistant  from  the  straight. 
To  prove  that  a  single  point  of  the  join  gives  a  perpendicular 
equal  to  those  f  rom  its  ends,  he  shows  would  be  sufficient. 

We  make  the  assumption  of  Clavius  when,  to  draw  a 
straight  line,  we  use  a  ruler  and  pencil.  Saccheri  shows 
this  is  dependent  on  the  "hypothesis  of  right  angle" ;  on  the 
other  hypotheses  the  equidistant  is  curved ;  in  obtuse,  con- 
vex  to  the  given  straight;  in  acute,  concave. 

The  most  beautiful  theorem  of  geometry  is  Euclid,  III, 
31 :  The  angle  in  a  semicircle  is  a  right  angle.  But  Sac- 
cheri's  Proposition  18  is:  According  as  an  angle  inscribed 
in  a  semicircle  is  right,  obtuse  or  acute,  the  hypothesis  of 
right,  obtuse  or  acute  angle  is  true. 

1  Kings  iii.  5  says:  "In  Gibeon  the  Lord  appeared  to 


ix 


Solomon  in  a  dream  by  night:  and  God  said,  Ask  what  I 
shall  give  thee." 

Then  says  Dante  of  his  asking, 

"  'Twas  not  to  know  the  number  In  which  are 


Or  if  in  semicircle  can  be  made 
Triangle  so  that  it  have  no  right  angle."i 

—Par.,  C.  XIII,  97-102. 

Now  what  could  Saccheri  think?  Let  us  remember  the 
authority  of  EucHd,  the  just  reverence  for  his  Elements. 
He  founds  geometry  upon  certain  assumptions  on  which  the 
whole  of  the  reasoning  rests.  No  aUernative  was  presumed 
possible.  In  1731,  two  years  before  Saccheri's  publication, 
in  a  book  with  almost  the  same  title  A  Defense  of  Euclid's 
Elements,  Edmund  Stone,  F.R.S.,  calls  it  "a  work  whose 
propositions  have  such  an  admirable  connection  and  de- 
pendence,  whose  demonstrations  are  so  convincing,  elegant 
and  perspicuous,  that  it  is  beyond  the  skill  of  man  to  con- 
trive  better.  This  is  the  happy  Empire  wherein  Truth  has 
had  an  uninterrupted  reign  for  upward  of  two  thousand 
years,  in  profound  peace." 

In  The  Wonderful  Century,  Alf  red  Russel  Wallace  says, 
speaking  of  all  time  bef ore  the  seventeenth  century :  "Then 
going  backward,  we  can  find  nothing  of  the  first  rank  except 
Euclid's  wonderful  system  of  geometry,  perhaps  the  most 
remarkable  product  of  the  earliest  civilizations." 

Says  Prof .  Alfred  Baker  of  the  University  of  Toronto : 
"Of  the  perfection  of  EucHd  (B.  C.  290)  as  a  scientific 
treatise,  of  the  marvel  that  such  a  work  could  have  been 
produced  two  thousand  years  ago,  I  shall  not  here  delay 
to  speak." 

Says  CHfford:  "This  book  has  been  for  nearly  twenty- 
two  centuries  the  encouragement  and  guide  of  that  scien- 
tific  thought  which  is  one  thing  with  the  progress  of  man 
from  a  worse  to  a  better  state." 

1  "O  se  del  mezzo  cerchio  far  si  puote 

Triangol  si,  ch'  un  retto  non  avesse." 


Philip  Kelland,  Cambridge  Senior  Wrangler,  says:  "It 
is  certain  that  from  its  completeness,  uniformity  and  fault- 
lessness,  from  its  arrangement  and  progressive  character, 
and  from  the  universal  adoption  of  the  completest  and  best 
line  of  argument,  EucHd's  Elements  stands  preeminently  at 
the  head  of  all  human  productions.  In  no  science,  in  no 
department  of  knowledge,  has  anything  appeared  Hke  this 
work.  For  upward  of  two  thousand  years  it  has  commanded 
the  admiration  of  mankind." 

For  two  millenniums  EucHd's  axioms  and  postulates  re- 
mained  undoubted.  Before  Saccheri,  no  one  had  even  for 
a  moment  thought  of  contradicting  any  of  them.  But  Sac- 
cheri  set  forth  two  propositions,  each  a  flat  contradiction 
of  EucHd's  most  famous  postulate,  the  fifth  (Heath),  then 
called  Axiom  13  (Clavius),  afterward  Axiom  12  (Simson), 
and  Axiom  11  (John  Bolyai).  These  two  new  monsters, 
hypotheses  of  his  own  creating,  he  attacked. 

EucHd  from  the  beginning  builds  consciously  upon  the 
assumption:  Two  points  determine  a  straight.  He  also 
uses  the  Archimedes  assumption,  and  the  assumption  that 
every  straight,  besides  being  unbounded,  divides  the  plane 
into  two  parts,  is  open,  infinite.  With  the  aid  of  these 
assumptions,  Saccheri  disposes  of  his  first  monster,  the 
hypothesis  anguli  obtusi. 

It  was  not  real  death,  however,  but  a  magician's  trance 
that  lasted  more  than  a  century.  FinaHy  freed  from  the 
spell  by  the  trumpet  caU  of  genius,  it  arises,  a  benignant 
fairy,  and  to-day  gives  us  Pure  Spherics,  deduced  from  a 
set  of  assumptions  which  give  no  paraHels,  no  similar  figures, 
but  double  the  value  of  much  of  our  plane  geometry  by 
interpreting  it  as  spherics  also,  by  showing  how  it  holds  good 
as  spherics,  can  be  read  off  as  spherics  and  used  as  spherics 
also.  Thus  drafts  from  EucHd  have  become  payable  twice 
over. 

But  apart  from  his  Postulate  V,  aH  EucHd's  assump- 
tions,  conscious  and  unconscious,  fail  Saccheri  in  his  lepgthy 
battle,  "diuturnum  proelium  adversus  hypothesin  anguli 
acuti" 


So  arises  the  first  non-Euclidean  geometry. 

In  his  book,  Euclid's  Parallel  PostulateJ^  Dr.  Withers  is 
another  who  yields  to  the  temptation  to  suppose  revealed 
Saccheri's  emotional  life,  where,  on  page  119,  he  says: 

"We  have  seen  that  the  former  assumption  [acuti]  wor- 
ried  Saccheri  very  profoundly  in  his  heroic  efforts  to  Vindi- 
cate  Euclid.'  It  was  not  a  logical  but  a  psychological  or 
experiential  difficulty  which  caused  Saccheri  to  reject  the 
logical  conclusions  to  which  his  own  labors  clearly  and 
inevitably  pointed;  and  it  was  certainly  the  same  sort  of 
difficulty  which  caused  the  immediate  rejection,  by  himself 
and  by  subsequent  mathematicians,  of  the  assumption  [ob- 
tusi]," 

Saccheri's  volume  is  divided  into  two  Books.  The  first, 
Propp.  I-XXXIX,  pp.  1-101,  we  give  entire.  The  second,  pp. 
102-142,  we  omit.  It  considers  two  of  Euclid's  definitions, 
Eu.  V.  def.  6,  now  numbered  5 ;  and  Eu.  VI.  def.  5,  now 
omitted.  This  "Liber  Secundus"  is  a  defense  of  the  pro- 
f  ound  treatment  of  proportion  in  Euclid's  Book  V.  It  shows 
again  Saccheri's  wisdom,  penetration  and  modernity. 

Remember  Perry  saying:  "I  wasted  much  precious  time 
of  my  lif e  on  the  fifth  book  of  Euclid,"  and  then  the  dictum  of 
the  great  Cayley,  "There  is  hardly  anything  in  mathematics 
more  beautiful  than  his  wondrous  fifth  book."  For  my  own 
part,  nothing  ever  better  repaid  study. 

In  this  reproduction,  the  original  pages  are  identified  by 
their  numbers  in  square  brackets.  Saccheri's  "Indicis  loco" 
refers  to  these.  For  quick  orientation,  however,  a  "Table 
of  Contents"  has  been  added,  giving  the  location  of  any 
particular  proposition,  scholion,  corollary,  etc,  by  the  page- 
numbers  of  this  reprint. 

Misprints  of  the  original  edition,  both  those  noticed  by 
Saccheri's  printer  and  those  overlooked,  have  been  corrected 
without  further  comment. 

It  is  a  piece  of  inestimable  good  fortune  that  the  page 
proofs  have  been  read  by  one  of  the  foremost  classical 
scholars  in  America,  Dr.  M.  W.  Humphreys,  who  says: 
2  The  Open  Court  Ptiblishing  Company,  1908. 


"The  Latin  is  almost  classical,  and  is  remarkably  clear.  The 
superiority  of  the  Latinity  over  that  of  the  New  Testament 
Vulgate  is  very  marked." 

How  budded  into  the  world  the  concepts  which  were  to 
make  of  the  Euclidean  geometry,  consecrated  by  the  tradi- 
tions  of  all  the  ages,  only  a  special  case,  a  species  of  a  genus, 
must  be  of  etemal  interest  in  the  history  of  thought,  and  no 
translation  can  suffice  without  the  original. 

For  the  constructive  part  of  Saccheri's  work,  the  first 
seventy  pages,  through  Proposition  32,  all  connoisseurs  have 
enthusiastically  expressed  their  admiration,  their  delight  in 
its  elegance,  its  exquisite  artistic  finish. 

G.  B.  H. 
Greeley,  Col.,  November,  1919. 


INTRODUCTION. 

Giovanni  Girolamo  Saccheri  was  bom  at  San  Remo  in 
the  night  between  the  4th  and  5th  of  September,  1667,  as 
the  authority  on  his  Hfe,  the  late  Alberto  Pascal  tells  us. 

He  was  notably  precocious. 

March  24,  1685,  he  entered  the  Jesuit  order.  Toward 
1690  he  terminated  the  period  of  novitiate  at  Genoa  and 
was  sent  by  his  superiors  to  the  Collegio  di  Brera  in  Milan, 
to  teach  grammar  and  at  the  same  time  to  study  philosophy 
and  theology.  There  the  reading  of  the  Elements  of  Euclid 
was  recommended  to  him  by  the  professor  of  mathematics, 
the  Jesuit  father  Tommaso  Ceva,  from  whose  brother  Gio- 
vanni  the  theorem  Ceva  is  named. 

In  1694  Saccheri  was  commanded  to  teach  philosophy 
and  polemic  theology  in  the  Collegio  dei  Gesuiti  of  Turin. 
In  1697  he  was  sent  to  Pavia.  Says  Pascal,  "Fruit  of  these 
three  years  of  philosophic  teaching  was  a  little  book  which 
well  merits  to  be  better  known:  perhaps  its  extreme  rarity 
has  contributed  to  this  obHvion,  even  since  Giovanni  Vailati, 
in  1903,  brought  to  Hght  its  superlative  merit." 

The  few  brief  pages  of  Vailati,  who  died  in  1909,  furnish 
the  only  help  on  this  lihretto,  an  opera  dimenticata  first  ap- 
pearing  with  what  he  caHs  the  titolo  abbastanza  enigmatico : 

Logica  demonstrativa,  quam  una  cum  Thesibus  ex  tota 
Philosophia  decerptis,  defendendam  proponit  Joannes  Fran- 
ciscus  Casalette  Graveriarum  Comes  sub  auspiciis  Regiae 
Celsitudinis  Victorii  Amedei  II.  Sabaudiae  Ducis,  Pede- 
montium  Principis,  Cypri  Regis,  etc. 


Augustae  Taurinorum  Typis  Joannis  Baptistae  Zappatae 
1697.     Superiorum  permissu. 

(In  16^  pp.  xii-287.) 

Thus,  in  this  first  edition,  the  name  of  the  author  does 
not  appear.  Taking  advantage  of  the  examination  of  Count 
Gravere,  then  his  student,  Saccheri,  with  the  theses,  pub- 
lished  his  course  in  logic,  letting  it  appear  as  if  the  count's. 

The  only  existing  copy  is  in  the  Biblioteca  Nazionale 
of  Milan  (Colloc.  B.  X.  4854). 

On  it  is  written: 

Auctore  P're.  Hyeronymo  Saccherio  Societatis  Jesu, 
and  below: 

Ex  Bihlioth'^  Collegii  Brayd'"  Soc'*^  Jesu.  /wj.[criptus] 
CataV'^. 

Saccheri,  astute  and  prudent,  had  his  reasons  for  issuing 
this  three-year  child  of  his  genius  under  the  count's  cloak, 
Then  as  professor  he  changed  subjects  and  residence,  and 
only  four  years  afterward  did  the  book  appear  with  his 
name. 

The  first  issue  Saccheri  never  mentions.  The  second 
edition,  so  called,  he  refers  to  repeatedly  and  insistently. 
It  differs  from  the  first  by  some  suppressions,  especially  in 
the  preface,  but  no  thought  has  been  added  during  these 
four  years  of  waiting. 

Its  title  is: 

Logica  demonstrativa  auctore  Hieronymo  Saccherio 
Societatis  Jesu,  olim  in  Collegio  Taurinensi  eiusdem  So' 
cietatis  Philosophiae,  ac  Theologiae  Polemicae,  nunc  in 
Archigymnasio  Ticinensi  Publico  Matheseos  Professore. 

Jllustriss.  Domino  D.  Philippo  Archinto  Sacr.  Rom.  Imp. 
Comiti,  Marchioni  Patronae,  Comit.  Trainati,  Domino  Er- 
bae  et  Terrar.  adiacen.  Plebis  Ticini,  et  Condom.  Alhizati, 
ac  Reg.  Duc.  Senatori  etc. 

Ticini  Regii.    MDCCI. 

Typis  Haeredum  Caroli  Francisci  Magrii  Impressorum 
Civit.  Superiorum  permissu. 

(In  8^  pp.  vi-167.) 


At  Colognc  in  1735  appeared  a  third  edition  after  the 
author^s  death  and  the  publication  of  his  Euclides  vindicatus 
in  1733. 

Its  title  is: 

Logica  demonstrativa,  Theologicis,  Philosophicis  et  Ma- 
thematicis  Disciplinis  accommodata;  Auctore  R.  P.  Hiero- 
nymo  Saccherio,  Societatis  Jesu,  olim  in  Collegio  Taurin- 
ensi  eiusdem  Societatis  Philosophiae  ac  Theologiae  Pole- 
micae;  nunc  in  Archi-Gymnasio  Ticinensi  publico  Mathe- 
seos  professore. 

Augustae  Ubiorum,  sumtu  Henrici  Noethen,  Bibliopolae, 
in  pladea  vulgo  dicta  unter  Helmschldger  sub  insigni  capitis 
aurei.    MDCCXXXV. 

(In  8^  pp.  vi-162.) 

The  editor  terminates  a  laudatory  preface  with  the 
epigram : 

"Si  tua,  Saccheri,  ingenio 
par  penna  fuisset, 
aetas  ostendat  vix  tibi 
nostra  parem." 

In  the  Stadtbibliothek  of  Cologne  (Augusta  Ubiorum) 
is  the  only  existing  copy  of  this  posthumous  edition. 

In  his  preface  our  author  says :  "Quattuor  in  partes  logi- 
cam  nostram,  cum  Aristotele,  dividimus.  Prima  docebit 
regulas  rectae  argumentationis ;  secunda  tradet  methodum 
tenendam  in  cognitionibus  scientificis ;  tertia  sternit  viam 
ad  cognitiones  opinativas ;  quarta  fallacias  detegit." 

The  scholastic  logic  undergoes  a  critical  elaboration 
which  takes  the  form  of  a  series  of  demonstrations  based 
upon  postulates  and  definitions  and  interconnected  in  a  way 
analogous  to  the  method  of  geometers.^ 

In  the  same  prelude  mention  is  made  of  what  he  judges 
new  and  important  contributions  to  the  ordinary  treatment 
of  logic. 

*  "Severa  illa  mcthodo  quae  primis  principiis  vix  parcit  nihilve 
non  clarum,  non  evidens,  non  indubitatum,  admittit— Ea  quam  dixi 
gcometriae  severitas  quae  nihil  indemonstratum  recipiat."    Ibid. 


Says  Heath:  "Miirs  account  of  the  true  distinction  be- 
tween  real  and  nominal  definitions  had  been  fully  antici- 
pated  by  Saccheri." 

In  his  Logica  demonstrativa  Saccheri  lays  down  the 
clear  distinction  between  what  he  calls  definitiones  quid 
nominis  or  nominales,  and  definitiones  quid  rei  or  reales, 
namely,  that  the  former  are  only  intended  to  explain  the 
meaning  that  is  to  be  attached  to  a  given  term,  whereas  the 
latter,  besides  declaring  the  meaning  of  a  word,  affirm  at 
the  same  time  the  existence  of  the  thing  defined  or,  in 
geometry,  the  possibiHty  of  constructing  it.  The  definitio 
quid  nominis  becomes  a  definitio  quid  rei  "by  means  of  a 
postulate,  or  when  we  come  to  the  question  whether  the 
thing  exists  and  it  is  answered  affirmatively."^ 

Definitiones  quid  nominis  are  in  themselves  quite  arbi- 
trary,  and  neither  require  nor  are  capable  of  proof ;  they 
are  merely  provisional,  and  are  only  intended  to  be  turned 
as  quickly  as  possible  into  definitiones  quid  rei,  either 

1.  by  means  of  a  postulate  in  which  it  is  asserted  or 
conceded  that  what  is  defined  exists  or  can  be  constructed, 
e.  g.,  in  the  case  of  straight  lines  and  circles,  to  which 
Euclid's  first  three  postulates  refer,  or 

2.  by  means  of  a  demonstration  reducing  the  construc- 
tion  of  the  figure  defined  to  the  successive  carrying-out  of 
a  certain  number  of  those  elementary  constructions,  the 
possibiHty  of  which  is  postulated.  Thus  definitiones  quid  rei 
are  in  general  obtained  as  the  result  of  a  series  of  demon- 
strations. 

Saccheri  gives  as  an  instance  the  construction  of  a  square 
in  EucHd  I.  46. 

Suppose  that  it  is  objected  that  EucHd  had  no  right  to 
define  a  square,  as  he  does  at  the  beginning  of  the  Book, 
when  it  was  not  certain  that  such  a  figure  exists;  the  ob- 
jection,  he  says,  could  only  have  force  if,  before  proving 
and  making  the  construction,  EucHd  had  assumed  the  af  ore- 

2  "Definitio  quid  nominis  nata  est  evadere  definitio  quid  rei  per 
postulatum  vel  dum  venitur  ad  quaestionem  an  est  et  respondetur 
affirmative."    Ihid. 


said  figure  as  given.  That  Euclid  is  not  guilty  of  this  error 
is  clear  f  rom  the  f  act  that  he  never  presupposes  the  existence 
of  the  square  as  defined  until  after  I.  46. 

Confusion  between  the  nominal  and  the  real  definition 
as  thus  described,  i.  e.,  the  use  of  the  former  in  demon- 
stration  before  it  has  been  turned  into  the  latter  by  the 
necessary  proof  that  the  thing  defined  exists,  is,  according 
to  Saccheri,  one  of  the  most  fruitful  sources  of  illusory 
demonstration,  and  the  danger  is  greater  in  proportion  to 
the  "complexity"  of  the  definition,  i.  e.,  the  number  of  vari- 
ety  of  the  attributes  belonging  to  the  thing  defined.  For 
the  greater  is  the  possibility  that  there  may  be  among  the 
attributes  some  that  are  incompatihle,  i.  e.,  the  simultaneous 
presence  of  which  in  a  given  figure  can  be  proved,  by  means 
of  other  postulates,  etc,  forming  part  of  the  basis  of  the 
science,  to  be  impossible. 

This  signal  anticipation  of  MilFs  famous  distinction 
would  alone  justify  the  only  known  protagonist  of  the 
Logica  demonstrativa  hitherto,  Vailati,  in  saying  of  Sac- 
cheri:  "This  gives  him  the  right  to  an  eminent  place  in  the 
history  of  modern  logic." 

But  in  additional  elaboration  Saccheri  broadens  the  mat- 
ter,  clearly  recognizing  the  more  general  question  relative 
to  the  necessity  of  excluding  the  possible  existence  of  in- 
compatibility  among  the  fundamental  postulates  made  the 
basis  of  a  demonstrative  science;  and  not  merely  their 
directly  contradicting  one  another,  but  whether  the  falsity 
of  one  of  them  could  be  proved  by  means  of  the  others, 
a  thing  not  directly  recognizable. 

These  questions,  far  from  having  grown  old,  are  ac- 
quiring  an  ever  greater  importance  with  the  accentuation 
of  the  modern  tendency  to  regard  as  the  function  of  mathe- 
matics,  the  development,  logically  coherent,  of  the  conse- 
quences  flowing  from  a  given  system  of  premises,  whether 
or  no  these  be  susceptible  of  a  direct  interpretation  or  ex- 
perimental  verification. 

Since  actually,  in  this  case,  the  postulates  assume  the 
character  of  simple  hypotheses  subject  only  to  the  condition 

ziz 


of  being  mutually  compatihle,  that  is,  of  neither  directly 
nor  indirectly  contradicting  one  another,  the  question  rela- 
tive  to  the  means  of  ascertaining  whether  such  compati- 
bility  really  exists,  ceases  to  be,  in  Vailati's  phrase,  a  pure 
question  de  luxe,  upon  its  solution  having  come  to  depend 
the  legitimacy  and  even  the  possibiHty  of  assuming  a  given 
system  of  hypotheses  as  basis  of  a  demonstrative  science. 

How  high  the  merit  of  having  been  far  the  first  to  en- 
visage  this  difficult  matter  and  to  have  proffered  an  analysis 
of  the  various  forms  of  fallacy  to  which  its  non-recognition 
may  give  rise! 

Precisely  to  such  subject  is  dedicated  the  final  chapter 
of  the  Logica  demonstrativa.  And  so  ultra-modern  and  yet 
unfinished  is  this  whole  question  here  raised  and  entered 
upon  first,  that  it  beckons  with  rising  interest  to  mathema- 
ticians  and  philosophers  toward  this  Httle  book  so  near  to 
vanishing  unrecognized  from  the  earth. 

"Huc  usque  de  fallaciis  communiter  observatis.  Duas 
adhuc  superaddemus  nec  eas  ut  opinor  parvi  momenti. 
Hanc  'fallaciam  complexi'  appeUo,  iHam  'dupHcis  defini- 
tionis'  seu  'hypothesis' "  (p.  256  of  Ist  ed.). 

The  fallacy  of  "complex  definitions,"  such  as  attribute 
to  the  thing  defined  the  simultaneous  possession  of  diverse 
properties,  as  for  example  BorelH's  of  "parallel"  the  prop- 
erty  of  being  a  straight  line  and  that  of  being  also  the  locus 
of  points  of  a  plane  equidistant  from  another  given  straight, 
consists  in  supposing  that  such  definitions  can  be  adopted 
unchecked  in  the  demonstrations,  without  the  compatibility 
of  the  properties  themselves  having  first  been  ascertained. 

Obviously,  in  case  such  compatibility  is  lacking,  in  case 
the  existence  of  an  object  possessing  simultaneously  the 
properties  in  question  can  be  proved  impossible  (by  means 
of  the  other  hypotheses  anteriorly  postulated  as  basis  of 
the  demonstrative  science  under  discussion),  any  argumen- 
tation  among  whose  premises  figure  such  definitions  com- 
bined  with  the  aforesaid  hypotheses,  ceases  to  have  value, 
being  based  on  contradictory  premises. 

The  forms  of  illusory  reasoning  examined  and  probed 


by  Saccheri  under  the  name  of  fallaciae  duplicis  hypothesis 
are  precisely  those  which  consist  in  believing  that  conse- 
quences  worth  considering  can  be  deduced  from  systems  of 
hypotheses  incompatible  with  one  another  (to  wit,  such 
that  among  them  are  some  whose  negation  can  be  deduced 
from  the  others)  ;  and  of  such  he  passes  in  review  various 
types,  beginning  with  the  simplest,  namely,  that  of  a  syllo- 
gism  whose  premises  directly  contradict  one  another,  and 
going  on  to  the  more  complicated  cases  in  which  the  contra- 
diction  can  be  revealed  only  by  the  successive  development 
of  the  consequences  of  the  system  of  hypotheses,  or  postu- 
lates,  assumed  as  basis  of  the  entire  matter. 

In  the  investigation  of  the  independence  of  postulates  the 
method  consists  in  finding  a  case  or  a  particular  interpre- 
tation  in  which  the  proposition  we  wish  to  prove  not  dedu- 
cible  f  rom  others  given,  ceases  to  be  true  while  all  the  others 
remain  true.  If  such  is  found,  we  conclude  that  the  propo- 
sition  cannot  be  deduced  f  rom  these  others,  else  it  would  be 
true  whenever  they  were. 

This  use  and  construction  of  examples  to  show  the 
independence  of  a  certain  proposition  from  others  given 
has  of  late  assumed  the  importance  of  an  ordinary  and 
indispensable  procedure  in  the  strictly  rigorous  elaboration 
of  any  deductive  theory  (in  America,  Robert  L.  Moore, 
Huntington,  Veblen,  and  others).  But  Saccheri  was  the 
first  to  use  this  procedure,  and  constructs  his  example  with- 
out  leaving  the  field  of  formal  logic.  If  his  treatment  of 
his  "hypothesis  of  acute  angle"  is  another  case,  it  is  the  most 
marvelous  in  the  world. 

In  his  Opus  de  proportionibus  (Hb.  V,  prop.  201),^* 
Cardan  (bom  at  Pavia,  1501),  to  prove  that  two  sides  of 
a  certain  triangle  he  has  occasion  to  consider  are  greater 
than  a  certain  arc  of  a  circle  comprised  between  them,  adopts 
a  pecuHar  reasoning  and  vaunts  himself  of  this  singular 
procedure  as  an  extraordinary  discovery  of  his  own: 

^Cardani  Opera,  Lugduni  [Lyons],  1663,  t  IV,  p.  579,  published 
by  Spon  in  ten  volumes,  folio. 


"Hanc  propositionem  non  scripsi,  quod  esset  magni  mo- 
menti,  sed  propter  modum  probandi. 

"Si  enim  respicis,  ex  uno  opposito  (scilicet  quod  peri- 
pheria  circuli  sit  major  trianguli  lateribus)  ostendo,  demon- 
stratione  non  ducente  ad  inconveniens  sed  simplici,  quod 
ipsa  peripheria  minor  est  trianguU  lateribus. 

"Et  hoc  nunquam  fuit  factum  ab  aliquo,  immo  videtur 
plane  impossibile,  et  est  res  admirabihor  quae  inventa  sit  ab 
orbe  condito,  scilicet  ostendere  aliquod  ex  suo  opposito, 
demonstratione  non  ducente  ad  impossibile,  et  ita  ut  non 
possit  demonstrari  ea  demonstratione  nisi  per  illud  sup- 
positum  quod  est  contrarium  conclusioni,  velut  si  quis  de- 
monstraret  quod  Socrates  est  albus  quia  est  niger,  et  non 
possit  demonstrare  aliter;  et  ideo  est  longe  majus  Chrysip- 
paeo  Syllogismo."* 

But  that  Cardan  had  been  anticipated  in  this  mode  of 
deduction  is  twice  noted  by  C.  Clavius  (1537-1612),  once 
apropos  of  a  demonstration  given  by  Theodosius  of  TripoH 
(Sphaericorum,  hb.  I.  prop.  12),  in  proof  of  the  theorem 
that  two  circles  on  the  same  sphere  cannot  bisect  one  another 
unless  they  are  great  circles: 

"Hic  vides  mirabile  arg^imentandi  modus  quod  ex  eo 
quod  dicitur  C  non  esse  centrum  sphaerae  demonstratum 
est,  demonstratione  affirmativa,  C  esse  centrum  sphaerae. 
Quo  modo  argumentandi  etiam  usus  est  EucHdes  (IX.  12.) 
et  Cardanus  De  proportionibus  (V.  prop.  201)"; 

and  again  in  a  schoHum  on  Eu.  IX.  12,  in  his  Euclidis  ele- 
mentorum  lihri  XV.    Roma.    8*^.     1574. 

EucHd's  proof  is  a  characteristic  example  of  this  logical 
procedure,  this  type  of  demonstration. 

*  "And  this  has  never  been  done  by  any  one :  nay,  it  seems 
clearly  impossible,  and  is  the  most  wonderful  thing  that  has  been 
devised  since  the  creation  of  the  world,  namely,  to  prove  something 
from  its  opposite,  the  demonstration  not  leading  to  an  impossibiHty, 
and  in  such  a  way  that  it  could  not  be  proved  by  that  demonstration 
except  by  that  being  supposed  which  is  contrary  to  the  conclusion, 
just  as  if  one  were  to  prove  that  Socrates  is  white  because  he  is 
black  and  could  not  prove  it  in  any  other  way;  and  for  that  reason 
it  is  far  greater  than  the  Chrysippaean  Syllogism." 

xxii 


EUCLID'S  ELEMENTS,  BOOK  IX,  PROPOSITION  12. 

//  there  be  how  many  numbers  soever  in  continued  propor- 
tion  from  unity :  Then  whatever  prime  numbers  measure 
the  last,  the  same  will  also  measure  that  next  after  the 
unit,  ^ 

Let  there  be  as  many  numbers  as  we  please,  A,  B,  C,  D 
continual  proportionals  from  unity;  I  say  whatever  prime 
numbers  measure  D  will  measure  A  also. 

Unity      A         B         C         D 
For  example,        1  4         16        64       256 

E        H         G         F 
2  8         32       128 

For  let  some  prime  number  E  measure  D;  I  say  E 
measures  A. 

For  suppose  it  does  not. 

Now  E  is  prime,  and  a  prime  number  is  prime  to  any  it 
does  not  measure  [VII.  29]  ;  therefore  E,  A  are  prime  to 
one  another. 

And  since  E  measures  D,  let  it  measure  it  by  the  units 
in  F;  therefore  E  multiplying  F  produces  D. 

Again,  since  A  measures  D  by  the  units  in  C  [IX.  11  and 
Porism],"^  therefore  A  multiplying  C  produces  D.  But  E 
has  also  by  multiplying  F  made  D ;  whence  the  product  of 
A,  C  is  equal  to  the  product  of  E,  F. 

Therefore,  as  A  to  E,  so  is  F  to  C  [VII.  19]. 

But  A,  E  are  prime;  primes  are  also  least  [VII.  21],* 
and  the  least  measure  those  which  have  the  same  ratio  the 
same  number  of  times,  the  antecedent  the  antecedent  and 
the  consequent  the  consequent  [VII.  20]  ;  therefore  E  meas- 
ures  C. 

Let  it  measure  it  by  G ;  then  E  multiplying  G  produces 
C.  But  A  has  also  by  multiplying  B  made  C  [IX.  11  and 
Porism] . 

8  X'"+"  =  X"  X". 

«"Numbers  prime  to  one  another  are  the  least  of  those  which 
have  the  same  ratio  as  they." 

xxiU 


Therefore  the  product  of  A,  B  is  equal  to  the  product 
of  E,  G. 

Whence,  as  A  to  E,  so  is  G  to  B  [VII.  19]. 

But  A,  E  are  prime;  primes  are  also  least  [VII.  21], 
and  do  equally  measure  those  that  have  the  same  ratio  as 
they,  the  antecedent  the  antecedent  and  the  consequent  the 
consequent  [VII.  20]  : 

Wherefore  E  measures  B. 

Let  it  measure  it  by  H ;  then  E  multiplying  H  produces 
B.  But  A  has  also  by  multiplying  itself  made  B  [IX.  8]  ; 
therefore  the  product  of  E,  H  is  equal  to  the  product  of 
A  into  itself. 

Therefore,  as  E  to  A,  so  is  A  to  H  [VII.  19]. 

But  A,  E  are  prime ;  primes  are  also  least  [VII.  21],  and 
the  least  measure  those  which  have  the  same  ratio  the  same 
number  of  times,  the  antecedent  the  antecedent  and  the 
consequent  the  consequent  [VII.  20]  ;  therefore  E  measures 
A,  as  antecedent  antecedent.     Q.  E.  D. 

The  comment  of  Clavius  is: 

"Est  autem  res  admirabiHs  huius  propositionis  demon- 
stratio.  Nam  ex  eo  quod  b  dicatur  non  metiri  ipsum  a, 
ostendit,  demonstratione  affirmativa,  b  ipsum  a  metiri,  quod 
videtur  fieri  non  posse.  Nam  si  quis  demonstrare  instituat 
Socratem  esse  album,  ex  eo  quod  non  est  albus,  paradoxum 
aliquid  et  inopinatum  in  medio  videatur  afiferre:  cui  tamen 
non  absimile  quid  factum  hic  est  in  numeris  ab  EucHde,  et 
in  aHis  nonnulHs  propositionibus  quae  sequuntur." 

Now  the  edition  of  Clavius  was  the  EucHd  recommended 
by  T.  Ceva  to  Saccheri. 

This  recondite  legerdemain  of  logic,  so  striking  to  Cardan 
and  Clavius,  seized  with  a  more  permanent  fascination  the 
adroit  mind  of  the  subtle  Saccheri.  It  is  the  dominant  note 
of  the  Logica  demonstrativa,  and  thence  adventures  the 
quest  of  the  holy  grail,  EucHd's  ParaHel  Postulate. 

This  type  of  reasoning  consists  in  assuming  as  hypothesis 
the  falsity  of  the  very  proposition  to  be  proved,  and  in  show- 
ing  how  also  when  taking  this  hypothesis  as  point  of  de- 


parture,  none  the  less  do  we  likewise  arrive  at  the  con- 
clusion  that  the  proposition  in  question  is  true. 

In  this  process  we  reach  nothing  absurd  or  false,  but 
thus  proceeding  we  attain  the  very  proposition  which  was 
to  be  proved,  so  that  in  this  way  it  shows  itself  as  a  con- 
sequence  of  its  own  negation/ 

In  his  preface,  Saccheri,  enumerating  the  parts  of  his 
work  in  which  he  beHeves  himself  to  have  made  new  and 
important  contributions  to  the  ordinary  treatment  of  logic, 
gives  first  place  to  Chapter  11  of  Part  I,  devoted  precisely 
to  this  type  of  reasoning.  He  might  still  claim  the  merit 
of  being  not  only  the  first  but  the  only  one  to  employ  this 
method  in  a  systematic  treatise  on  logic.  To  have  applied 
it  to  the  elaboration  of  the  rules  of  the  scholastic  logic, 
Saccheri  regards  as  one  of  the  most  important  ameHorations 
introduced  by  him  in  the  treatment  of  the  subject,  and  for 
us  it  is  highly  significant  to  note  how  the  greatest  advantage 
inherent  in  this  innovation  of  his  consists  for  him  in  his 
being  able  by  this  means  to  render  his  exposition  indepen- 
dent  of  the  assumption  of  a  certain  postulate  he  beHeves 
indispensable  if  the  ordinary  treatment  be  followed. 

Hence  there  is  an  exact  correspondence  between  the 
use  he  makes  in  his  Logica  of  this  demonstrative  procedure, 
and  the  use  he  afterward  attempted  to  make  of  it  in  his 
Euclides  vindicatus,  aiming  to  obviate  the  necessity  of  as- 
suming  the  Parallel  Postulate. 

Vailati,  of  whose  few  precious  pages  we  avail  ourselves, 
points  out  that  Saccheri  tells  how  already  in  his  youth  he 
arrived  at  the  idea  that  the  characteristic  property  of  the 
most  fundamental  propositions,  in  every  demonstrative  sci- 
ence,  was  precisely  their  being  indemonstrable  except  by 
recourse  to  this  IX — 12  type  of  argumentation  (see  Eu. 
vind.,  pp.  99-100),  and  then  adds: 

"It  was  in  hopes  of  reaching  in  this  way  a  proof  of  the 
Parallel  Postulate,  namely,  deducing  it  from  the  very  hy- 
pothesis  of  its  falsity,  that  Saccheri  pushed  on  in  the  in- 

'  "Sumam  contradictoriam  propositionum  demonstrandarum  ex 
coque,  ostensive  ac  directe,  propositum  eliciam."    Log.  dem.,  p.  130. 


vestigation  of  the  consequences  flowing  from  the  other 
two  alternative  hypotheses  to  which  the  negation  of  the 
Parallel  Postulate  gave  rise,  attaining  thus  results  fitting 
to  carry  on  in  their  sequence  to  a  discovery  far  more  im- 
portant  than  what  he  had  in  mind  to  reach,  namely  to  the 
discovery  of  a  whoUy  new  geometry  of  which  the  old  is  only 
a  simple  particular  case. 

"In  this  regard,  his  position  is  not  unworthy  to  be  com- 
pared  with  that  of  his  great  fellow-countryman  Columbus, 
who,  precisely  in  hopes  of  being  able  to  reach  by  a  new  way 
regions  already  known,  was  led  to  the  discovery  of  a  new 
continent." 

Twisting  Vailati's  fine  comparison,  Saccheri's  file  of 
Indians  tumed  out  to  be  Columbians    (Americans). 

ReaHzing  its  importance  as  the  coconut  out  of  whose 
eyes  the  palm  was  to  shoot  up,  which  rises  high  above  the 
flat  and  circumscribed  old  world,  let  us  further  look  at  the 
little  Logica. 

Its  Chapter  9,  of  Part  I,  deduces  the  ordinary  rules  of 
the  scholastic  logic,  relative  to  the  conversion  of  propositions 
and  to  the  construction  of  the  various  kinds  of  syllogisms, 
by  a  procedure  imitating  that  followed  by  geometers  in  their 
treatises,  and  notes  how  in  such  procedure  it  is  often  neces- 
sary  to  have  recourse  to  the  assumption  that,  given  any 
term,  it  is  always  possible  to  find  other  terms  which  are 
not  coextensive  with  it  nor  with  its  negation.^  It  proposes 
then  to  re-elaborate  the  same  subject  following  a  method 
other  and  more  refined  (aliam  nobiliorem  viam)  with  which 
there  is  no  need  of  using  the  mentioned  postulate. 

This  aim  is  attained  by  taking  as  point  of  departure 
those  among  the  propositions  antecedently  proved  in  whose 
demonstration  no  use  has  been  made  of  the  assumption  in 
question,  and  by  deducing  f  rom  these,  recourse  being  had  to 
the  IX — 12  form  of  argument,  the  remaining  propositions, 
which  before  had  been  obtained  by  using  the  assumption 
thus  eliminated. 

8  "Postulatur  non  omnes  terminos  esse  pertinentes  mutua  sequela 
aut  repugnantia,  sed  quosdam  esse  inferiores  et  superiores,  quosdam 
etiam  impertinentes."    Log.  dem.,  p.  30. 


Here,  for  instance,  is  the  demonstration  in  nobiliorem 
viam  of  the  noted  rule  of  the  scholastic  logic  according  to 
which  in  syllogisms  of  the  so-called  First  Figure  (in  syllo- 
gisms,  namely,  in  which  the  subject  and  the  predicate  of  the 
conclusion  enter  respectively  as  subject  and  predicate  also 
in  the  premises),  the  premise  in  which  appears  the  subject 
of  the  conclusion  cannot  be  negative :  In  prima  figura  minor 
non  potest  esse  negativa. 

Taking  the  simplest  particular  case,  it  is  to  be  proved 
that  from  the  two  propositions : 

Every  A  is  a  B, 
No  C  is  an  A, 

can  be  inferred  no  general  proposition  (affirmative  or  nega- 
tive)  having  C  as  subject  and  B  as  predicate. 

Proposing  to  demonstrate  this  rule  by  means  of  only  the 
syllogism  Barhara,  which  has  its  two  premises  universal  and 
affirmative,  Saccheri  first  observes  that  his  aim  would  be 
attained  if,  for  each  of  the  different  forms  of  syllogisms 
with  negative  minor  premise  constructible  in  the  first  figure, 
he  could  succeed  in  finding  examples  (that  is,  could  choose 
such  particular  meanings  for  the  terms  entering)  for  which 
the  two  premises  being  true,  the  conclusion  was  false: 

"Si  quispiam  syllogismus  taliter  constructus  non  recte 
concludit,  nullus  alius  simiHter  constructus  vi  formae  con- 
cludet"  (Logica  dem.,  p.  130). 

[Is  this  Italy,  1697,  or  America,  1919?] 

For  example,  to  prove  that,  from  the  two  premises 

Every  A  is  a  B, 
No  C  is  an  A, 

we  cannot  deduce  the  conclusion 

No  C  is  a  B. 

Attribute  to  the  terms  A,  B,  C,  respectively  the  three 
f ollowing  significations : 
A  =  syllogism  of  the  first  figure,  having  the  two  premises 

universal  and  affirmative; 
B  =  a  valid  syllogism ; 


C  =  syllogism  of  the  first  figure  having    one  premise  neg- 
ative. 
The  two  premises 

Every  A  is  a  B, 
No  C  is  an  A, 

will  then  become  the  two  following  propositions : 

1.  Every  syllogism  of  the  first  figure  having  the  two 
premises  universal  and  affirmative  is  a  vahd  syllo- 
gism: 

2.  No  syllogism  of  the  first  figure  having  one  premise 
negative,  is  a  syllogism  of  the  first  figure  having  the 
two  premises  universal  and  affirmative. 

Now  these  two  premises  both  being  true,  we  must  either 
admit  as  true  the  conclusion: 

No  C  is  a  B 

(that  is,  No  syllogism  of  the  first  figure  having  one  premise 
negative  is  a  valid  syllogism),  or  else  concede  that  the  mean- 
ings  we  have  given  to  the  terms  A,  B,  C  of  the  syllogism 
whose  validity  is  in  question,  render  true  its  two  premises 
and  false  its  conclusion. 

In  either  case  we  are  equally  forced  to  admit  that  the 
syllogism  in  question  is  not  valid. 

"Vel  concedis  vel  negas  consequentiam.  Si  concedis, 
habetur  intentum.  Si  negas,  conclusioni  dissentiens  post 
concessas  praemissas,  fateris  ipse  legitimum  non  esse  ex 
praemissis  eiusmodi  consequentiam  quod  intendebatur" 
(Logica  dem.,  p.  132). 

His  teaching  of  logic  ended,  his  course  published,  its 
weapon  of  predilection,  IX — 12,  left  in  his  powerful  hands, 
in  his  new  field,  mathematics,  what  heroic  adventure  was 
worthy  its  trenchant  edge? 

Sir  Henry  Savile,  in  his  Praelectiones  tresdecim  in  prin- 
cipium  Elementorum  Euclidis  hdbitae  1620,  Oxford,  4®, 
1621,  p.  140,  says:  "In  pulcherrimo  Geometriae  corpore 
duo  sunt  naevi." 

And  the  greatest  of  these  moles  is  the  eternal  Parallel 


Postulate.  Here  then  is  something  worthy  of  Saccheri's 
steel.     To  prove  it  from  its  own  denial! 

This  would  show  that  Euclid's  assumptions,  though  com- 
patible,  were  not  all  independent.  On  the  other  hand,  the 
independence  of  the  Parallel  Postulate  from  the  other  as- 
sumptions  would  be  established  if  it  were  shown  to  be  in- 
demonstrable  from  them  even  with  the  help  of  its  own  con- 
tradictory  opposite,  that  is,  even  by  means  of  Saccheri's 
darling  type,  IX — 12. 

To  get  this  negative  in  convenient  form,  Saccheri  uses 
for  it  an  equivalent,  employing  a  figure  found  in  his  Clavius, 
1574,  and  again  in  Giordano  Vitale  da  Bitonto,  in  his  Euclide 
restituto  overo  gli  antichi  elementi  geometrici  ristaurati,  e 
facilitati.  Libri  XV.  Roma.  fol.,  1680 ;  and  in  both  works  pre- 
cisely  in  discussion  of  this  very  matter.  The  figure  is  the 
isosceles  bi-rectangular  quadrilateral.  Its  other  two  angles 
are  equal.  To  assume  one  right  is,  with  Euclid's  other  as- 
sumptions,  equivalent  to  the  parallel  postulate,  whose  nega- 
tive  therefore  is  to  assume  one  oblique.  Armed  then  with 
this  form  of  its  denial  as  an  addition  to  the  other  Euclidean 
assumptions,  Saccheri  sallies  forth  to  the  fray,  steeled  for 
victory  or  def eat — but  not  for  the  wholly  unexpected  and  to 
him  inexplicable  compound  of  victory  and  defeat  which  he 
met. 

His  negation  breaks  into  two  equal  parts.  The  angle 
assumed  oblique  is  either  obtuse  or  acute.  If  it  be  obtuse, 
he  easily  achieves  his  accustomed  victory:  he  proves  the 
Parallel  Postulate.    If  it  be  acute,  this  twin  will  not  win. 

Why?     We  know.     Saccheri  never  did. 

Besides  the  Archimedes  assumption,  Euclid,  and  every 
one  else  for  more  than  a  century  after  Saccheri,  assumes 
that  the  straight  line  is  of  infinite  length.  These  assumptions 
nullify  the  possibility  of  a  pair  of  obtuse  angles  in  a  bi- 
rectangular  isosceles  quadrilateral,  and  to  that  extent  prove 
the  "hypothesis  of  right  angle,"  which  is  then  equivalent  to 
the  Parallel  Postulate.  But  they  are  no  obstacle  to  this  pair 
of  angles  being  acute. 

Had  there  been  some  other  unconscious  assumption  of 


Euclid's,  preventing  their  being  acute,  then  Saccheri  might 
well  have  declared  the  Parallel  Postulate  completely  demon- 
strated. 

But  there  is  none. 

Under  the  "hypothesis  of  acute  angle*'  the  chain  of 
beautiful  theorems  developed,  grew,  but  did  not  end. 

So  flowered  the  beauteous  body  of  a  new  geometry, 
mermaid-like,  the  latter  portions  somewhat  fishy,  but  oh! 
the  elegant  torso. 

Of  this  book  says  the  genius  Corrado  Segre:  "Never- 
theless  the  first  seventy  pages  (apart  from  a  few  isolated 
phrases),  up  to  Proposition  32  inclusive,  constitute  an  en- 
semble  of  logic  and  of  geometric  acumen  which  may  be 
called  perfect." 


EUCLIDES  VINDICATUS 


EUCLIDES 

AB  OMNI  N>£VO  VlNDlCATUSr 

S  I  VE 

CONATUS  GEOMETRICUS 

QUO    STABILIUNTUR 

Prima  ipla  univerfat  Geometria:  Principia. 
AU  C  T  OK  E 

HIERONYMO  SACCHERIO 

SOCIETATIS    JESU 

In  Ticinenfi  Univerfitate  Mathefeos  Profeflbre. 

OPUSCULUM 

EX^^"  SENATUI 

MEDIOLANENSI 

Ab  Audore  Dicatum. 

M  ED  1  O  L  A  N  I,  MDCCXXXIII. 

£x  Typograptiia  P&ult  Aaconit  Momaoi  •       Su^erkrum  ^ermiffir^ 


EUCLID 

FREED  OF  EVERY  FLECK 


OR 


A  GEOMETRIC  ENDEAVOR  IN  WHICH  ARE 

ESTABLISHED  THE  FOUNDATION 

PRINCIPLES  OF  UNIVERSAL 

GEOMETRY 


BY 

GIROLAMO  SACCHERI 

OF  THE  SOCIETY  OF  JESUS 

PROFISSOR  OF  MATHEMATICS  IN  THE  UNIVERSITY  OF  FAVIA. 


A  WORK  DBDICATED  TO 
THE  NOBLE  SENATE  OF 
MILAN  BY  THE  AUTHOR 


MILAN,  1733 

PAOLO  ANTONIO  MONTANO  SUPERIORUM   PERMISSU 


PROCEMIUM  AD  LECTOREM. 

Quanta  sit  Elementorum  Euclidis  praestantia,  ac  dig- 
nitas,  nemo  omnium,  qui  Mathematicas  disciplinas  no- 
verint,  ignorare  potest.  Lectissimos  hanc  in  rem  testes 
adhibeo  Archimedem,  Apollonium,  Theodosium,  aHosque 
pene  innumeros,  ad  haec  usque  nostra  tempora  rerum 
Mathematicarum  Scriptores,  qui  non  ahter  haec  EucHdis 
Elementa  usurpant,  nisi  ut  principia  jam  diu  stabiHta, 
ac  penitus  inconcussa.  Verum  tanta  haec  nominis  celebri- 
tas  vetare  non  potuit,  quin  multi  ex  Antiquis  pariter,  ac 
Recentioribus,  iique  Magni  Geometrae  naevos  quosdam 
a  se  deprehensos  censuerint  in  his  ipsis  pulcherrimis, 
nec  unquam  satis  laudatis  Elementis.  Tres  autem  hujus- 
modi  naevos  designant,  quos  statini  subnecto. 

Primus  respicit  definitionem  parallelarum,  et  sub  ea 
Axioma,  quod  apud  Clavium  est  decimumtertium  Libri 
primi,  ubi  EucHdes  sic  pronunciat :  Si  in  duas  rectas  lineas, 
in  eodem  plano  existentes  recta  incidens  linea  duos  ad 
easdem  partes  internos  angulos  minores  duobus  rectis 
cum  eisdem  efficiat,  duae  illae  rectae  lineae  ad  eas  partes 
in  infinifum  protractae  inter  se  mutuo  incident.  Porro 
nemo  est,  qui  dubitet  de  veritate  expositi  Pronunciati; 
sed  in  eo  unice  EucHdem  accusant,  quod  nomine  Axio- 
matis  usus  fuerit,  quasi  nempe  ex  soHs  terminis  rite  per- 
spectis  sibi  ipsi  faceret  fidem.  Inde  autem  non  pauci 
(retenta  caeteroquin  EucHdaea  parallelarum  definitione) 


PREFACE  TO  THE  READER. 

Of  all  who  have  learned  mathematics,  none  can  fail 
to  know  how  great  is  the  excellence  and  worth  of  Euclid's 
Elements.  As  erudite  witnesses  here  I  summon  Archi- 
medes,  Apollonius,  Theodosius,  and  others  almost  in- 
numerable,  writers  on  mathematics  even  to  our  times. 
who  use  Euchd's  Elements  as  foundation  long  estabHshed 
and  wholly  unshaken.  But  this  so  great  celebrity  has  not 
prevented  many,  ancients  as  well  as  moderns,  and  among 
them  distinguished  geometers,  maintaining  they  had 
found  certain  blemishes  in  these  most  beauteous  nor  ever 
sufificiently  praised  Elements.  Three  such  flecks  they  des- 
ignate,  which  now  I  name. 

The  first  pertains  to  the  definition  of  parallels  and 
with  it  the  axiom  which  in  Clavius  is  the  thirteenth  of 
the  First  Book,  where  EucHd  says : 

//  a  straight  line  falling  on  two  straight  lines,  lying 
in  the  same  plane,  make  with  them  tzvo  internal  angles 
toward  the  same  parts  less  than  two  right  angles,  these 
two  straight  lines  infinitely  produced  toward  those  parts 
ivill  meet  each  other. 

No  one  doubts  the  truth  of  this  proposition ;  but  solely 
they  accuse  Euchd  as  to  it,  because  he  has  used  for  it  the 
name  axiom,  as  if  obviously  from  the  right  understanding 
of  its  terms  alone  came  conviction.  Whence  not  a  few 
(withal  retaining  EucHd's  definition  of  parallels)   have 


illius  demonstrationem  aggressi  sunt  ex  iis  solis  Proposi- 
tionibus  Libri  primi  Euclidaei,  quae  praecedunt  vigesi- 
mam  nonam,  ad  quam  scilicet  usui  esse  incipit  contro- 
versum  Pronunciatum.  M 

Sed  rursum;  quoniam  antiquorum  in  hanc  rem  cona- 
tus  visi  non  sunt  adamussim  scopum  attingere;  factum 
idcirco  est,  ut  multi  proximiorum  temporum  eximii  Geo- 
metrae,  idem  pensum  aggressi,  necessariam  censuerint 
novam  quandam  parallelarum  definitionem.  Itaque ;  cum 
Euclides  parallelas  rectas  lineas  definiat,  quae  in  eodem 
plano  existentes,  si  in  utranque  partem  in  infinitum  pro- 
ducantur,  nunquam  inter  se  mutuo  incidunt;  postremis 
expositae  definitionis  vocibus  has  ahas  substituunt:  Sem- 
per  inter  se  aequidistant ;  adeo  ut  nempe  singulae  perpen- 
diculares  ab  uno  quoHbet  unius  illarum  puncto  ad  alteram 
demissae  aequales  inter  se  sint. 

At  nova  rursum  hinc  oritur  scissura.  Nam  aliqui,  et 
ii  sane  acutiores,  demonstrare  conantur  parallelas  rectas 
Hneas  prout  sic  definitas,  unde  utique  gradum  faciant  ad 
demonstrandum  sub  ipsis  EucHdaeis  vocibus  controver- 
sum  Pronunciatum,  cui  nimirum  ab  ea  vigesima  nona 
Libri  primi  EucHdaei  (paucuHs  quibusdam  exceptis)  uni- 
versa  innititur  Geometria.  AHi  vero  (non  sine  magno 
in  rigidam  Logicam  peccato)  eas  tales  rectas  Hneas  paral- 
lelas,  nimirum  aequidistantes,  assumunt  tanquam  datas, 
ut  inde  gradum  faciant  ad  reHqua  demonstranda. 

Et  haec  quidem  satis  sunt  ad  praemonendum  Lecto- 
rem  super  iis,  quae  materiam  exhibebunt  Libro  priori 
hujus  mei  OpuscuH :  Nam  uberior  praedictorum  omnium 
expHcatio  habebitur  in  SchoHis  post  Prop.  vigesimam 
primam  enunciati  Libri,  quem  dividam  in  duas  veluti 
partes.  In  priore  imitabor  antiquiores  iHos  Geometras, 
nihil  propterea  soHicitus  de  natura,  aut  nomine  iUius 
Hneae,  quae  omnibus  suis  punctis  a  quadam  supposita 
recta  Hnea  aequidistet :  Sed  unice  in  id  incumbam,  ut  con- 

6 


attempted  its  demonstration  from  those  propositions  of 
Euclid's  First  Book  alone  which  precede  the  twenty- 
ninth,  wherein  begins  the  use  of  the  controverted  propo- 
sition.  M 

But  again,  since  the  endeavors  of  the  ancients  in  this 
matter  do  not  seem  to  attain  the  goal,  so  it  has  happened 
that  many  distinguished  geometers  of  ensuing  times,  at- 
tacking  the  same  idea,  have  thought  necessary  a  new 
definition  of  parallels.  Thus,  while  Euclid  defines  paral- 
lels  as  straight  Hnes  lying  in  the  same  plane,  which,  if 
infinitely  produced  toward  hoth  sides,  nowhere  meet,  they 
substitute  for  the  last  words  of  the  given  definition  these 
others:  always  equidistant  from  each  other;  so  that  all 
perpendiculars  from  any  points  on  one  of  them  let  fall 
upon  the  other  are  equal  to  one  another. 

But  again  here  arises  a  new  fissure.  For  some,  and 
these  surely  the  keenest,  endeavor  to  demonstrate  the 
existence  of  parallel  straight  Hnes  as  so  defined,  whence 
they  go  up  to  the  proof  of  the  debated  proposition  as 
stated  in  EucHd's  terms,  upon  which  truly  from  that 
twenty-ninth  of  EucHd's  First  Book  (with  some  very 
few  exceptions)  all  geometry  rests.  But  others  (not 
without  gross  sin  against  rigorous  logic)  assume  such 
paraHel  straight  Hnes,  forsooth  equidistant,  as  if  given, 
that  thence  they  may  go  up  to  what  remains  to  be  proved. 

And  this  is  enough  to  indicate  to  the  reader  what  will 
be  the  material  of  the  First  Book  of  this  work  of  mine : 
for  a  more  complete  expHcation  of  ah  that  has  been  said 
wiU  be  given  in  the  schoHa  after  the  twenty-first  propo- 
sition  of  this  Book. 

I  divide  this  Book  into  two  parts.  In  the  First  Part 
I  wiH  imitate  the  antique  geometers,  and  not  trouble  my- 
self  about  the  nature  or  the  name  of  that  Hne  which  at 
aH  its  points  is  equidistant  from  a  certain  Hne  supposed 
straight;  but  merely  undertake  without  any  petitio  prin- 


troversum  Euclidaeum  Axioma  citra  omnem  petitionem 
principii  clare  demonstrem;  nunquam  idcirco  adhibens 
ex  ipsis  prioribus  Libri  primi  Euclidaei  Propositionibus, 
non  modo  vigesimam  septimam,  aut  vigesimam  octavam, 
sed  nec  ipsas  quidem  decimam  sextam,  aut  decimam  septi- 
mam,  nisi  ubi  clare  agatur  de  triangulo  omni  [xi]  ex  parte 
circumscripto.  Tum  in  posteriore  parte,  ad  novam  ejus- 
dem  Axiomatis  confirmationem  demonstrabo  non  nisi  rec- 
tam  lineam  esse  posse,  quae  omnibus  suis  punctis  a  qua- 
dam  supposita  recta  linea  aequidistet.  Horum  autem 
occasione  prima  ipsa  universae  Geometriae  Principia 
rigido  examini  subjicienda  hic  esse  nullus  est,  qui  non 
videat. 

Transeo  ad  alios  duos  naevos  Euclidi  objectos.  Prior 
respicit  definitionem  sextam  Libri  quinti  super  aeque  pro- 
portionaHbus :  Posterior  Definitionem  quintam  Libri  sexti 
super  compositione  rationum.  Hic  autem  erit  secundi  mei 
Libri  unicus  scopus,  ut  dilucide  expHcem  praefatas  EucH- 
daeas  Definitiones,  simulque  ostendam  non  aequo  jure 
hac  in  parte  EucHdis  nomen  vexatum  fuisse. 

Prodest  tamen  rursum  praemonere,  demonstratum  a 
me  iri  hac  occasione  ununi  quoddam  Axioma,  quod  tutis- 
sime  per  omnem  Geometriam  versetur,  sine  indigentia 
ihius  Posfulafi,  sub  nomine  Axiomatis  ab  interpretibus 
(ut  reor)  intrusi,  cujus  usus  incipit  ad  18.  quinti.  [xii] 


cipii  clearly  to  demonstrate  the  disputed  Euclidean  axiom. 
Tlierefore  never  will  I  use  from  those  prior  propositions 
of  EucHd's  First  Book,  not  merely  the  twenty-seventh 
or  the  twenty-eighth,  but  not  even  the  sixteenth  or  the 
seventeenth,  except  where  clearly  it  is  question  of  a  tri- 
angle  every  [^i]  way  restricted. 

Then  in  the  Second  Part  for  a  new  confirmation  of 
the  same  axiom,  I  shall  demonstrate  that  the  Hne  which 
at  all  its  points  is  equidistant  from  an  assumed  straight 
line  can  only  be  a  straight  hne.  But  every  one  sees  that 
on  this  occasion  the  very  first  principles  of  all  geometry 
are  to  be  subjected  to  a  rigid  examination. 

I  go  on  to  the  other  two  blemishes  charged  against 
Euchd.  The  first  pertains  to  the  sixth  definition  of  the 
Fifth  Book  about  proportionals ;  the  second  to  the  fifth 
definition  of  the  Sixth  Book  about  the  composition  of 
ratios.  It  will  be  the  sole  aim  of  my  Second  Book  to 
clearly  expound  the  Euchdean  definitions  mentioned,  and 
at  the  same  time  to  show  that  EucHd's  fame  is  here  un- 
justly  attacked. 

Yet  again  it  is  weh  to  state  that  on  this  occasion  I 
shall  prove  a  certain  axiom  that  may  safely  be  appHed 
throughout  the  whole  of  geometry,  without  need  of  that 
postulate,  put  in  (as  I  beheve)  by  commentators  under 
the  name  of  axiom,  whose  use  begins  at  the  eighteenth 
proposition  of  the  Fifth  Book.  [xii] 


INDICIS  LOCO 

ADDENDA  CENSEO,  QUAE   SEQUUNTUR. 

1.  In  L  et  11.  Propos.  Lib.  primi  duo  jaciun- 
tur  principia,  ex  quibus  in  III.  et  IV.  demon- 
stratur,  angulos  interiores  ad  rectam  jungentem 
extremitates  aequalium  perpendiculorum,  quae 
ex  duobus  punctis  alterius  rectae,  veluti  basis, 
versus  easdem  partes  (in  eodem  plano)  erigan- 
tur,  non  modo  fore  inter  se  aequales,  sed  prae- 
terea  aut  rectos,  aut  obtusos,  aut  acutos,  prout 
illa  jungens  aequalis  fuerit,  aut  minor,  aut  major 
praedicta  basi :  Atque  ita  vicissim.  apag.  1 

2.  Hinc  sumitur  occasio  secernendi  tres  di- 
versas  hypotheses,  unam  anguH  recti,  alteram 
obtusi,  tertiam  acuti :  circa  quas  in  V.  VI.  et 
VII.  demonstratur,  unam  quamHbet  harum  hypo- 
thesium  fore  semper  unice  veram,  si  nimirum 
depraehendatur  vera  in  uno  quoHbet  casu  parti- 
culari.  apag.S 

3.  Tum  vero;  post  interpositas  tres  aHas 
necessarias  Propositiones ;  demonstratur  in  XI. 
XII.  ac  XIII.  universaHs  veritas  noti  Axiomatis, 
respectu  habito  ad  priores  duas  hypotheses, 
unam  anguH  recti,  et  alteram  obtusi ;  ac  tandem 
in  XIV.  ostenditur  absoluta  falsitas  hypothesis 


IN  PLACE  OF  AN  INDEX 

SHOULD  BE  ADDED,  I  THINK,  WHAT  FOLLOWS. 

1.  In  Propp.  I.  and  11.  of  the  First  Book 
two  principles  are  established,  from  which  in 
Propp.  III.  and  IV.  is  proved,  that  interior  an- 
gles  at  the  straight  joining  the  extremities  of 
equal  perpendiculars  erected  toward  the  same 
parts  (in  the  same  plane)  from  two  points  of 
another  straight,  as  base,  not  merely  are  equal 
to  each  other,  but  besides  are  either  right  or 
obtuse  or  acute  according  as  that  join  is  equal 
to,  or  less,  or  greater  than  the  aforesaid  base: 

and  inversely.  From  page  1  on. 

2.  Hence  occasion  is  taken  to  distinguish 
three  different  hypotheses,  one  of  right  angle, 
another  of  obtuse,  a  third  of  acute :  about  which 
in  Propp.  V.,  VI.,  and  VII.  is  proved,  that  any 
one  of  these  hypotheses  is  always  alone  true  if 

it  is  f  ound  true  in  any  one  particular  case.  From  page  5  on. 

3.  Then  after  the  interposition  of  three  other 
necessary  propositions,  is  proved  in  Propp.  XI., 
XIL,  and  XIII. ,  the  universal  truth  of  the  fa- 
mous  axiom,  respect  being  had  to  the  first  two 
hypotheses,  one  of  right  angle,  and  the  other 
of  obtuse;  and  at  length  in  P.  XIV.  is  shown 
the  absolute  falsity  of  the  hypothesis  of  obtuse 


anguli  obtusi.  Atque  hinc  incipit  diuturnum 
proelium  adversus  hypothesin  anguH  acuti,  quae 
sola  renuit  veritatem  illius  Axiomatis.  [xiii]       apag.  10 

4.  Itaque  in  XV.  ac  XVI.  demonstratur  sta- 
biHtum  iri  hypotheses  aut  anguH  recti,  aut  ob- 
tusi,  aut  acuti,  ex  quoHbet  triangulo  rectiHneo, 
cujus  tres  simul  anguH  aequales  sint,  aut  ma- 
jores,  aut  minores  duobus  rectis ;  ac  similiter  ex 
quoHbet  quadrilatero  rectiHneo,  cujus  quatuor 
simul  anguH  aequales  sint,  aut  majores,  aut  mi- 

nores  quatuor  rectis.  a  pag.  20 

5.  Sequuntur  quinque  aHae  Propositiones, 
in  quibus  demonstrantur  aHa  indicia  pro  secer- 
nenda  vera  hypothesi  a  falsis.  a  pag.  23 

6.  Accedunt  quatuor  principaHa  SchoHa;  in 
quorum  postremo  exhibetur  figura  quaedam  geo- 
metrica,  ad  quam  fortasse  respexit  EucHdes,  ut 
suum  iUud  Pronunciatum  assumeret  tanquam 
per  se  notum.  In  tribus  prioribus  ostenditur 
non  valuisse  ad  intentum  praecedentes  insignium 
Geometrarum  conatus.  Sed  quia  controversum 
Axioma  exactissime  demonstratur  ex  duabus 
praesuppositis  rectis  Hneis  aequidistantibus ; 
monet  ibi  Auctor  contineri  in  eo  praesupposito 
manifestam  petitionem  Principii.  Quod  si  pro- 
vocari  hic  veHt  ad  communem  persuasionem, 
atque  item  exploratissimam  praxim;  rursum 
monet  provocari  non  debere  ad  experientiam, 
quae  respiciat  puncta  infinita,  cum  satis  esse 
possit  unica  experientia  uni  cuivis  puncto  affixa. 
Quo  loco  tres  ab  ipso  afferuntur  invictissimae 
Demonstrationes  Physico-Geometricae.  apag.  29 

7.  Supersunt  duodecim  ahae  Proposi-  [xiv] 
tiones,  quae  primae  Parti  hujus  Libri  finem  im- 
ponunt.  Non  expono  particularia  assumpta,  quia 


angle.  And  here  begins  a  lengthy  battle  against 
the  hypothesis  of  acute  angle,  which  alone  op- 
poses  the  truth  of  that  axiom.  [xiii]        From  page  10  on. 

4.  And  so  in  Propp.  XV.  and  XVI.  is  proved 
that  the  hypothesis  either  of  right  angle,  or  ob- 
tuse,  or  acute  is  established  f  rom  any  rectiHneal 
triangle  whose  three  angles  together  are  equal  to, 
or  greater,  or  less  than  two  right  angles ;  and 
in  Hke  way  from  any  rectiHneal  quadrilateral, 
whose  four  angles  are  together  equal  to,  or 
greater,  or  less  than  four  right  angles.    From  page  20  on. 

5.  Five  other  propositions  foUow,  in  which 
are  proved  other  indications  for  distinguishing 

the  true  hypothesis  from  the  false.  From  page  23  on 

6.  Now  come  four  fundamental  schoHa.  In 
the  last  is  exhibited  a  certain  geometric  figure, 
of  which  EucHd  perhaps  thought,  in  order  that 
his  proposition  might  assume  self-evidence.  In 
the  preceding  three  is  shown  that  the  prior  en- 
deavors  of  distinguished  geometers  have  not 
reached  their  aim.  Since  however  the  debated 
axiom  can  be  exactly  proved  from  two  straight 
Hnes  presupposed  equidistant,  the  author  here 
shows  a  manifest  petitio  principii  to  be  con- 
tained  in  that  presupposition.  If  one  wishes 
here  to  appeal  to  common  persuasion,  and  surest 
experience,  again  he  shows  appeal  should  not 
be  taken  to  an  experience  involving  an  infinity 
of  points,  when  a  single  experiment  pertaining 
to  any  one  point  can  sufiice.  In  this  place  are  set 
forth  by  him  three  invincible  physico-geometric 
demonstrations.  From  page  29  on. 

7.  To  the  end  of  the  First  Part  of  this  Book 
there  remain  twelve  other  propositions.  [xiv] 
I  do  not  state  the  particular  assumptions,  be- 


nimis  implexa.  Solum  dico  ibi  tandem  manifestae 
falsitatis  redargui  inimicam  hypothesim  anguli 
acuti,  utpote  quae  duas  rectas  agnoscere  deberet, 
quae  in  uno  eodemque  puncto  commune  recipe- 
rent  in  eodem  plano  perpendiculum :  Quod  qui- 
dem  naturae  lineae  rectae  repugnans  esse  de- 
monstratur  per  quinque  Lemmata,  in  quibus 
concluduntur  quinque  principaHa  Geometriae 
Axiomata,  quae  respiciunt  Hneam  rectam,  ac  cir- 
culum,  cum  suis  correlativis  Postulatis.  a  pag.  43 

8.  Secunda  pars  continet  sex  Propositiones. 
Ibi  autem;  post  expensam  (juxta  hypothesim 
anguH  acuti)  naturam  iHius  Hneae,  quae  omni- 
bus  suis  punctis  a  quadam  praesupposita  recta 
Hnea  aequidistet;  multis  modis  ostenditur,  eam 
fore  aequalem  contrapositae  basi,  unde  infertur 
praenunciatae  hypothesis  certissima  falsitas. 
Quare  tandem  in  ultima  Propos.  quae  est 
XXXIX.  exactissime  demonstratur  celebre  iHud 
EucHdaeum  Axioma,  cui  nempe  (ut  omnes  sci- 

unt)  universa  Geometria  innititur.  apag.SJ 

9.  Secundus  Liber  digeri  commode  non 
potuit  in  Propositiones,  etiamsi  locis  opportunis 
plura  intermista  sint  utiHssima  Theoremata,  ac 
Problemata.  Meretur  nihilominus  expresse  no- 
tari  unum  quoddam  Axioma,  cujus  ibi  demon- 
stratur  non  modo  veritas,  verum  etiam  univer- 
saHs  utiHtas  fxv]  pro  omni  Geometria,  sine  indi- 
gentia  alterius  parum  decori  Postulati,  quod  ab 
interpretibus  censeri  potest  intrusum  sub  nomine 
Axiomatis,  cujus  nempe  usus  incipit  ad  18.  quinti. 
Et  id  quidem  pro  prima  Parte  hujus  Libri,  in 

qua  vindicatur  Def.  sexta  quinti  EucHdaei.         apag.  102 


14 


cause  they  are  too  complex.  I  only  say  here  at 
length  I  have  disproved  the  hostile  hypothesis  of 
acute  angle  by  a  manifest  falsity,  since  it  must 
lead  to  the  recognition  of  two  straight  Hnes 
which  at  one  and  the  same  point  have  in  the 
same  plane  a  common  perpendicular.  That  this 
is  contrary  to  the  nature  of  the  straight  line  is 
proved  by  five  lemmas,  in  which  are  contained 
five  fundamental  axioms  relating  to  the  straight 
Hne  and  circle,  with  their  correlative  postul- 
lates.  From  page  43  on. 

8.  The  Second  Part  contains  six  proposi- 
tions.  Here,  after  investigating  the  nature  (as- 
suming  the  hypothesis  of  acute  angle)  of  that 
Hne  which  at  aH  its  points  is  equidistant  f  rom  an 
assumed  straight  Hne,  it  is  shown  in  many  ways 
that  it  equals  the  base  opposite,  whence  is  in- 
ferred  the  certain  falsity  of  the  aforesaid  hy- 
pothesis.  Wherefore  at  length  in  the  last  propo- 
sition,  P.  XXXIX.,  is  exactly  proved  that  famous 
axiom  of  EucHd,  upon  which  (as  everybody 
knows)  the  whole  of  geometry  rests.    From  page  87  on. 

9.  The  Second  Book  cannot  conveniently  be 
divided  into  propositions,  although  at  opportune 
places  are  intercalated  many  most  useful  theo- 
rems  and  problems.  Nevertheless  is  worthy  of 
express  mention  a  certain  axiom,  of  which  not 
merely  the  truth  is  there  demonstrated  but  also 
the  universal  utiHty  for  aH  geometry,  without 
need  of  the  other  inelegant  postulate  supposably 
inserted  by  commentators  under  the  name  of 
axiom,  whose  use  begins  at  Eu.  V.  18.  So 
much  for  the  First  Part  of  this  Book,  in  which 

is  defended  Eu.  V.  def.  6.  From  page  102  on. 


15 


10.  Tum  in  secunda  Parte;  praeter  nonnulla 
alia  opportune  addita,  ad  tuendas  reliquas  Defi- 
nitiones  ejusdem  Quinti  circa  magnitudines  pro- 
portionales;  demonstratur  priore  loco  (respectu 
habito  ad  magnitudines  commensurabiles)  quinta 
Definitio  Sexti,  etiamsi  recipi  ea  deberet  in  quid 
rei,  veluti  Axioma :  Sed  rursum  multis  exemplis, 
ex  ipso  Euclide  petitis,  ostenditur  nullius  de- 
monstrationis  indigam  eam  esse,  quia  Defini- 
tionem  puri  iiominis.  Atque  ita,  post  oppor- 
tunam  additam  Appendicem,  huic  Operi  finis 
imponitur.  apag.  132 


i6 


10.  Then  in  the  Second  Part,  besides  some 
other  things  opportunely  added  regarding  other 
definitions  of  Eu.  V  about  proportional  magni- 
tudes,  is  demonstrated  in  the  first  place  (with 
respect  to  commensurable  magnitudes)  Eu.  VI. 
def.  5,  even  if  it  ought  to  be  taken  in  qtnd  rei 
Hke  an  axiom.  But  on  the  contrary  is  shown  by 
many  examples  drawn  from  EucHd  himself  that 
this  needs  no  demonstration,  because  a  definition 
puri  noniinis.  And  so  after  an  Appendix  oppor- 
tunely  added,  an  end  is  put  to  this  work. 

From  page  132  on  {to  page  142] 


t7 


EUCLIDIS  AB  OMNI  NAEVO  VINDI- 

CATI 

LIBER  PRIMUS: 

IN  QUO  DEMONSTRATUR :  DUAS  QUASLIBET  IN  EODEM 
PLANO  EXISTENTES  RECTAS  LINEAS,  IN  QUAS  RECTA 
QUAEPIAM  INCIDENS  DUOS  AD  EASDEM  PARTES  IN- 
TERNOS  ANGULOS  EFFICIAT  DUOBUS  RECTIS  MINORES, 
AD  EAS  PARTES  ALIQUANDO  INVICEM  COITURAS^  SI  IN 
INFINITUM  PRODUCANTUR. 

PARS  PRIMA 

PROPOSITIO  I. 

Si  duae  aequales  rectae  (fig.  1.)  AC,BD,  aequales  ad  eas- 
dem  partes  efficiant  angulos  cum  recta  AB :  Dico 
angulos  ad  junctam  CD  aequales  invicem  fore. 

Demonstratur.  Jungantur  AD,  CB.  Tum  conside- 
rentur  triangula  CAB,  DBA.  Constat  (ex  quarta  primi) 
aequales  fore  bases  CB,  AD.  Deinde  considerentur  tri- 
angula  ACD,  BDC.  Constat  (ex  octava  primi)  aequales 
fore  angulos  ACD,  BDC.    Quod  erat  demonstrandum. 


i8 


EUCLID  FREED  OF  EVERY  FLECK. 

BOOK  I. 

IN  WHICH  is  proved:  any  two  coplanar  straight 
lines,  falling  upon  which  any  straight  makes 
toward  the  same  parts  two  internal  angles 
less  than  two  right  angles^  at  length  meet 
each  other  toward  those  PARTS^  if  infinitely 

PRODUCED. 


PART  I. 


PROPOSITION  L 

//  two  equal  straights  [sects]  (fig.  1)  AC,  BD,  make 
with  the  straight  AB  angles  equal  toward  the  same 
parts:  I  say  that  the  angles  at 
the  join  CD  will  be  mutually 
equal. 


Proof.  Join  AD,  CB.  Then 
consider  the  triangles  CAB,  DBA. 
It  follows  (Eu.  I.  4)  that  the  bases  pjg  j 

CB,  AD  will  be  equal. 

Then  consider  the  triangles  ACD,  BDC.     It  follows 
(Eu.  I.  8)  that  the  angles  ACD,  BDC  will  be  equal. 

Quod  erat  demonstrandum. 


PROPOSITIO  II. 

Manente  uniformi  quadrilatero  ABCD,  latera  AB,  CD, 
hifariam  dividantur  {fig.  2.)  in  punctis  M,  et  H.  Di- 
[2]  co  angidos  ad  junctam  MH  fore  hinc  inde  rectos. 

Demonstratur.  Jungantur  AH,  BH,  atque  item  CM, 
DM.  Quoniam  in  eo  quadrilatero  anguli  A,  et  B  positi 
sunt  aequales,  atque  item  (ex  praecedente)  aequales  sunt 
anguli  C,  et  D;  constat  ex  quarta  primi  (cum  alias  nota 
sit  aequalitas  laterum)  aequales  fore  in  triangulis  CAM, 
DBM,  bases  CM,  DM ;  atque  item,  in  triangulis  ACH, 
BDH,  bases  AH,  BH.  Quare ;  collatis  inter  se  triangulis 
CHM,  DHM,  ac  rursum  inter  se  triangulis  AMH,  BMH ; 
constabit  (ex  octava  primi)  aequales  invicem  fore,  atque 
ideo  rectos  angulos  hinc  inde  ad  puncta  M,  et  H.  Quod 
erat  demonstrandum. 

PROPOSITIO  III. 

Si  duae  aequales  rectae  (fig.  3.)  AC,  BD,  perpendictda- 
riter  insistant  cuivis  rectae  AB :  Dico  junctam  CD 
aequalem  fore,  aut  minorem,  aut  majorem  ipsa  AB, 
prout  anguli  ad  eandem  CD,  fuerint  aut  recti,  aut 
obtusi,  aut  acuti. 

Demonstratur  prima  pars.  Existente  recto  utroque 
angulo  C,  et  D;  sit,  si  fieri  potest,  alterutra  ipsarum,  ut 
DC,  major  altera  BA.    Sumatur  in  DC  portio  DK  aequa- 


Fig.  2. 


PROPOSmON  II. 
Retaining  the  uniform  quadrilateral  ABCD,  bisect  the 
sides  AB,  CD  (fig.  2)  in  the  points  M  and  H.  [2] 
/  say  the  angles  at  the  join 
MH  will  then  be  right. 

Proof.    Join  AH,  BH,  and 
likewise  CM,  DM. 

Because  in  this  quadrilateral 
the  angles  A  and  B  are  taken 
equal  and  likewise  (from  the 
preceding  proposition)  the  angles  C,  and  D  are  equal;  it 
follows  (Eu.  1.4)  (noting  the  equahty  of  the  sides)  that 
in  the  triangles  CAM,  DBM,  the  bases  CM,  DM  will  be 
equal;  and  Hkewise,  in  the  triangles  ACH,  BDH,  the 
bases  AH,  BH. 

Therefore;  comparing  the  triangles  CHM,  DHM, 
and  in  turn  the  triangles  AMH,  BMH ;  it  follows  (Eu. 
I.  8)  that  we  have  mutually  equal,  and  therefore  right, 
the  angles  at  the  points  M,  and  H. 

Quod  erat  demonstrandum. 

PROPOSITION  III. 
//  two  equal  straights  [sects]    {fig.  3)  AC,  BD,  stand 
perpendicular  to  any  straight  AB:  I  say  the  join 
CD  will  be  equal  to,  or 
less,  or  greater  than  AB, 
according  as  the  angles 
at  CD  are  right,  or  ob- 
tuse,  or  acute. 

Proof    of    the    First 
Part.    Each  angle  C,  and  D, 
being  right;   suppose,   if   it 
were  possible,  either  one  of  those,  as  DC,  greater  than 
the  other  BA. 


Fig.  3. 


lis  ipsi  BA,  jungaturque  AK.  Quoniam  igitur  super  BD 
perpendiculariter  insistunt  aequales  rectae  BA,  DK, 
aequales  erunt  (ex  prima  hujus)  anguli  BAK,  DKA.  Hoc 
autem  absurdum  est;  cum  angulus  BAK  sit  ex  construc- 
tione  minor  supposito  recto  BAC;  et  angulus  DKA  sit 
ex  constructione  externus,  atque  ideo  (ex  decimasexta 
primi)  major  interno,  et  opposito  DCA,  qui  supponitur 
rectus.  Non  ergo  alterutra  praedictarum  rectarum,  DC, 
BA,  est  altera  major,  dum  anguli  ad  junctam  CD  sint 
recti;  ac  propterea  aequales  invicem  sunt.  Quod  erat 
primo  loco  demonstrandum.  [3] 

Demonstratur  secunda  pars.  Si  autem  obtusi  fuerint 
anguli  ad  junctam  CD,  dividantur  bifariam  AB,  et  CD, 
in  punctis  M,  et  H,  jungaturque  MH.  Quoniam  ergo  su- 
per  recta  MH  perpendiculariter  insistunt  (ex  praece- 
dente)  duae  rectae  AM,  CH,  poniturque  ad  junctam  AC 
angulus  rectus  in  A,  non  erit  (ex  prima  hujus)  recta  CH 
aequaHs  ipsi  AM,  cum  desit  angulus  rectus  in  C.  Sed 
neque  erit  major:  caeterum  sumpta  in  HC  portione  KH 
aequah  ipsi  AM,  aequales  forent  (ex  prima  hujus)  anguH 
ad  junctam  AK.  Hoc  autem  absurdum  est,  ut  supra. 
Nam  angulus  MAK  est  minor  recto ;  et  angulus  HKA  est 
(ex  decimasexta  primi)  major  obtuso,  quaHs  supponitur 
internus,  et  oppositus  HCA.  Restat  igitur,  ut  CH,  dum 
anguH  ad  junctam  CD  ponantur  obtusi,  minor  sit  ipsa 
AM ;  ac  propterea  prioris  dupla  CD  minor  sit  posterioris 
dupla  AB.     Quod  erat  secundo  loco  demonstrandum. 

Demonstratur  tertia  pars.  Tandem  vero,  si  acuti  fue- 
rint  anguH  ad  junctam  CD,  ducta  pariformiter  (ex  prae- 
cedente)  perpendiculari  MH,  sic  proceditur.     Quoniam 


Take  in  DC  the  piece  DK  equal  to  BA,  and  join  AK. 
Since  therefore  on  BD  stand  perpendicular  the  equal 
straights  BA,  DK,  the  angles  BAK,  DKA  will  be  equal 
(P.  L).  But  this  is  absurd;  since  the  angle  BAK  is  by 
construction  less  than  the  assumed  right  angle  B AC ;  and 
the  angle  DKA  is  by  construction  external,  and  therefore 
(Eu.  I.  16)  greater  than  the  internal  and  opposite  DCA, 
which  is  supposed  right.  Therefore  neither  of  the  afore- 
said  straights,  DC,  BA,  is  greater  than  the  other,  whilst 
the  angles  at  the  join  CD  are  right;  and  therefore  they 
are  mutually  equal. 

Quod  erat  primo  loco  demonstrandum.  [3] 

Proof  of  the  Second  Part.  But  if  the  angles  at 
the  join  CD  are  obtuse,  bisect  AB,  and  CD,  in  the  points 
M,  and  H,  and  join  MH. 

Since  therefore  on  the  straight  MH  stand  perpendicu- 
lar  (P.  n.)  the  two  straights  AM,  CH,  and  at  the  join 
AC  is  a  right  angle  at  A,  the  straight  CH  will  not  be 
(P.  I.)  equal  to  this  AM,  since  a  right  angle  is  lacking 
at  C. 

But  neither  will  it  be  greater:  otherwise  in  HC  the 
piece  KH  being  assumed  equal  to  this  AM,  the  angles  at 
the  join  AK  will  be  (P.  I.)  equal. 

But  this  is  absurd,  as  above.  For  the  angle  MAK  is 
less  than  a  right;  and  the  angle  HKA  is  (Eu.  I.  16) 
greater  than  an  obtuse,  such  as  the  internal  and  opposite 
HCA  is  supposed. 

It  remains  therefore,  that  CH,  whilst  the  angles  at  the 
join  CD  are  taken  obtuse,  is  less  than  this  AM;  and 
therefore  CD  double  the  former  is  less  than  AB  double 
the  latter. 

Quod  erat  secundo  loco  demonstrandum. 

Proof  of  the  Third  Part.  Finally,  however,  if  the 
angles  at  the  join  are  acute,  MH  being  constructed  as  be- 
fore  perpendicular  (P.  II.),  we  proceed  thus.     Since  on 

«3 


super  recta  MH  perpendiculariter  insistunt  duae  rectae 
AM,  CH,  poniturque  ad  junctam  AC  angulus  rectus  in  A, 
non  erit  (ut  supra)  recta  CH  aequalis  ipsi  AM,  cum  desit 
angulus  rectus  in  C.  Sed  neque  erit  minor:  caeterum; 
si  in  HC  protracta  sumatur  HL  aequalis  ipsi  AM ;  aequa- 
les  forent  (ut  supra)  anguli  ad  junctam  AL.  Hoc  autem 
absurdum  est.  Nam  angulus  MAL  est  ex  constructione 
major  supposito  recto  MAC;  et  angulus  HLA  est  ex 
constructione  internus,  et  oppositus,  atque  ideo  minor 
(ex  decimasexta  primi)  externo  HCA,  qui  supponitur 
acutus.  Restat  igitur,  ut  CH,  dum  anguli  ad  junctam 
CD  sint  acuti,  major  sit  ipsa  AM,  atque  ideo  prioris  dupla 
CD  major  sit  posterioris  dupla  AB.  Quod  erat  tertio 
loco  demonstandum. 

Itaque  constat  junctam  CD  aequalem  fore,  aut  mino- 
[4]  rem,  aut  majorem  ipsa  AB,  prout  anguli  ad  eandem 
CD  fuerint  aut  recti,  aut  obtusi,  aut  acuti.  Quae  erant 
demonstranda. 

COROLLARIUM  L 

Hinc  in  omni  quadrilatero  continente  tres  quidem  an- 
gulos  rectos,  et  unum  obtusum,  aut  acutum,  latera  adja- 
centia  illi  angulo  non  recto  minora  sunt,  alterum  altero, 
lateribus  contrapositis,  si  ille  angulus  sit  obtusus,  majora 
autem,  si  sit  acutus.  Id  enim  demonstratum  jam  est  de 
latere  CH  relate  ad  contrapositum  latus  AM;  similique 
modo  ostenditur  de  latere  AC  relate  ad  contrapositum 
latus  MH.  Cum  enim  rectae  AC,  MH,  perpendiculares 
sint  ipsi  AM,  nequeunt  (ex  prima  hujus)  esse  invicem 
aequales,  propter  inaequales  angulos  ad  junctam  CH. 
Sed  neque  (in  hypothesi  anguH  obtusi  in  C)  potest  quae- 


the  straight  MH  stand  perpendicular  two  straights  AM, 
CH,  and  at  the  join  AC  is  a  right  angle  at  A,  the  straight 
CH  will  not  be  equal  to  this  AM  (as  above),  since  the 
angle  at  C  is  not  right.  But  neither  will  it  be  less: 
otherwise,  if  in  HC  produced  HL  is  taken  equal  to  this 
AM,  the  angles  at  the  join  AL  will  be  (as  above)  equal. 

But  this  is  absurd.  For  the  angle  MAL  is  by  con- 
struction  greater  than  the  assumed  right  MAC;  and  the 
angle  HLA  is  by  construction  internal,  and  opposite,  and 
therefore  less  than  (Eu.  L  16)  the  external  HCA,  which 
is  assumed  acute. 

It  remains  therefore,  that  CH,  whilst  the  angles  at 
the  join  CD  are  acute,  is  greater  than  this  AM,  and  there- 
fore  CD  the  double  of  the  former  is  greater  than  AB  the 
double  of  the  latter. 

Quod  erat  tertio  loco  demonstrandum. 

Therefore  it  is  estabHshed  that  the  join  CD  will  be 
equal  to,  or  less,  [4]  or  greater  than  this  AB,  according 
as  the  angles  at  the  same  CD  are  right,  or  obtuse,  or 
acute. 

Quae  erant  demonstranda. 

COROLLARY  L 

Hence  in  every  quadrilateral  containing  assuredly 
three  right  angles,  and  one  obtuse,  or  acute,  the  sides 
adjacent  to  this  oblique  angle  are  less  respectively  than 
the  opposite  sides  if  this  angle  is  obtuse,  but  greater  if 
it  is  acute. 

For  this  has  just  now  been  demonstrated  of  the  side 
CH  relatively  to  the  opposite  side  AM ;  in  the  same  way 
it  is  demonstrated  of  the  side  AC  relatively  to  the  oppo- 
site  side  MH.  For  since  the  straights  AC,  MH,  are 
perpendicular  to  this  AM,  they  cannot  (P.  L)  be  mutually 
equal,  on  account  of  the  unequal  angles  at  the  join  CH. 

But  neither  (in  the  hypothesis  of  an  obtuse  angle  at 

25 


dam  AN,  portio  ipsius  AC,  aequalis  esse  ipsi  MH,  qua 
nimirum  major  sit  praedicta  AC:  caeterum  (ex  eadem 
prima)  aequales  forent  anguli  ad  junctam  HN;  quod  est 
absurdum,  ut  supra.  Rursum  vero  (in  hypothesi  anguli 
acuti  in  eo  puncto  C)  si  veHs  quandam  AX,  sumptam  in 
AC  protracta,  aequalem  ipsi  MH,  qua  nimirum  minor 
sit  modo  dicta  AC;  jam  eodem  titulo  aequales  erunt  an- 
guH  ad  HX;  quod  utique  absurdum  itidem  est,  ut  supra. 
Restat  igitur,  ut  in  hypothesi  quidem  anguH  obtusi  in  eo 
puncto  C,  latus  AC  minus  sit  contraposito  latere  MH ;  in 
hypothesi  autem  anguH  acuti  sit  eodem  majus.  Quod 
erat  intentum. 

COROLLARIUM  IL 

Multo  autem  magis  erit  CH  major  portione  quaHbet 
ipsius  AM,  ut  puta  PM,  ad  quam  nempe  juncta  [5]  CP 
acutiorem  adhuc  angulum  efficiat  cum  ipso  CH  versus 
partes  puncti  H,  et  obtusum  (ex  decimasexta  primi) 
cum  ea  PM  versus  partes  puncti  M. 

COROLLARIUM  III. 

Rursum  constat  praedicta  omnia  aeque  procedere;. 
sive  assumpta  perpendicula  AC,  et  BD,  fuerint  certae 
cujusdam  apud  nos  longitudinis,  sive  sint,  aut  supponan- 
tur  infinite  parva.  Quod  quidem  notari  opportune  debet 
in  reHquis  sequentibus  Propositionibus. 

PROPOSITIO  IV. 

Vicissim  autem  (manenfe  figura  praecedentis  Proposi- 
tionis)  anguli  ad  junctam  CD  erunt  aut  recti,  aut 
obtusij  aut  acuti,  prout  recta  CD  aequalis  fuerit,  aut 
minor,  aut  major,  contraposita  AB. 


26 


C)  can  a  certain  AN,  a  piece  of  this  AC,  than  which 
certainly  the  aforesaid  AC  is  greater,  be  equal  to  this 
MH :  otherwise  (P.  I.)  the  angles  at  the  join  HN  would 
be  equal;  which  is  absurd,  as  above. 

Again  however  (in  the  hypothesis  of  an  acute  angle 
at  this  point  C),  if  you  take  a  certain  AX,  assumed  on 
AC  produced,  than  which  certainly  the  just  mentioned 
AC  is  less,  equal  to  this  MH ;  now  by  this  same  title  the 
angles  at  HX  will  be  equal;  which  assuredly  is  absurd 
in  the  same  way,  as  above. 

It  remains  therefore,  that  indeed  in  the  hypothesis  of 
an  obtuse  angle  at  this  point  C,  the  side  AC  is  less  than 
the  opposite  side  MH;  but  in  the  hypothesis  of  an  acute 
angle  is  greater  than  it. 

Quod  erat  intentum. 

COROLLARY  IL 
But  by  much  more  will  CH  be  greater  than  any  piece 
of  this  AM,  as  for  instance  PM,  since  of  course  the  join 
[5JCP  makes  an  angle  still  more  acute  with  this  CH 
toward  the  parts  of  the  point  H,  and  obtuse  (Eu.  I.  16) 
with  this  PM  toward  the  parts  of  the  point  M. 

COROLLARY  IIL 

Again  it  abides  that  all  things  aforesaid  equally  result, 
whether  the  assumed  perpendiculars  AC,  and  BD  are  of 
some  length  fbced  by  us,  or  are,  or  are  supposed  infini- 
tesimal. 

This  indeed  ought  opportunely  to  be  noted  in  remain- 
ing  subsequent  propositions. 

PROPOSITION  IV. 
But  inversely   (the  figure  of  the  preceding  proposition 
remaining)  the  angles  at  the  join  CD  will  be  right, 
or  obtuse,  or  acute,  according  as  the  straight  CD  is 
equalj  or  less,  or  greater  than  the  opposite  AB. 


Demonstratur.  Si  enim  recta  CD  aequalis  sit  contra- 
positae  AB,  et  nihilominus  anguli  ad  eandem  sint  aut  ob- 
tusi,  aut  acuti;  jam  ipsi  tales  anguli  eam  probabunt  (ex 
praecedente)  non  aequalem,  sed  minorem,  aut  majorem 
contraposita  AB;  quod  est  absurdum  contra  hypothesim. 
Idem  uniformiter  valet  circa  reHquos  casus.  Stat  igitur 
angulos  ad  junctam  CD  esse  aut  rectos,  aut  obtusos,  aut 
acutos,  prout  recta  CD  aequaHs  fuerit,  aut  minor,  aut  ma- 
jor  contraposita  AB.     Quod  erat  demonstrandum. 

DEFINITIONES. 

Quandoquidem  (ex  primahujus)  recta  jungens  extre- 
mitates  aequaHum  perpendiculorum  eidem  rectae  (quam 
vocabimus  basim)  insistentium,  aequales  ef-[6]ficit  an- 
gulos  cum  ipsis  perpendicuHs ;  tres  idcirco  distinguendae 
sunt  hypotheses  circa  speciem  horum  angulorum.  Et  pri- 
mam  quidem  appeHabo  hypothesim  anguH  recti ;  secun- 
dam  vero,  et  tertiam  appeHabo  hypothesim  anguH  obtusi, 
et  hypothesim  anguH  acuti. 

PROPOSITIO  V. 

Hypothesis  anguli  recti,  si  vel  in  uno  casu  est  vera,  sem- 
per  in  omni  casu  illa  sola  est  vera. 

Demonstratur.  Efficiat  juncta  CD  (fig.  4.)  angulos 
rectos  cum  duobus  quibusvis  aequaHbus  perpendicuHs  AC. 
BD,  uni  cuivis  AB  insistentibus.  Erit  CD  (ex  tertia  hu- 
jus)  aequaHs  ipsi  AB.  Sumantur  in  AC,  et  BD  protrac- 
tis  duae  CR,  DX,  aequales  ipsis  AC,  BD;  jungaturque 
RX.  Facile  ostendemus  junctam  RX  aequalem  fore  ipsi 
AB,  et  angulos  ad  eandem  rectos.     Et  primo  quidem  per 


28 


Proof.  For  if  the  straight  CD  is  equal  to  the  opposite 
AB,  and  nevertheless  the  angles  at  it  are  either  obtuse,  or 
acute;  now  these  such  angles  prove  it  (P.  III.)  not 
equal,  but  less,  or  greater  than  the  opposite  AB;  which 
is  absurd  against  the  hypothesis. 

The  same  uniformly  avails  in  regard  to  the  remain- 
ing  cases.  It  holds  therefore  that  the  angles  at  the  join 
CD  are  either  right,  or  obtuse,  or  acute,  according  as  the 
straight  CD  is  equal  to,  or  less,  or  greater  than  the  oppo- 
site  AB. 

Quod  erat  demonstrandum. 

DEFINITIONS. 
Since  (P.  I.)  the  straight  joining  the  extremities  of 
equal  perpendiculars  standing  upon  the  same  straight 
(which  we  call  base),  makes  equal  [6]  angles  with  these 
perpendiculars ;  therefore  there  are  three  hypotheses  to 
be  distinguished  according  to  the  species  of  these  angles. 
And  the  first  indeed  I  will  call  hypothesis  of  right  angle ; 
the  second  however,  and  the  third  I  will  call  hypothesis 
of  obtuse  angle,  and  hypothesis  of  acute  angle. 

PROPOSITION  V. 
//  even  in  a  single  case  the  hypothesis  of  right  angle 
is  true,  always  in  every  case  it  alone  is  true. 

Proof.     Let  the  join  CD  (fig.  4)     _ 
make  right  angles  with  any  two  per- 
pendiculars    AC,    BD,    standing   upon 
any  straight  AB. 

CD  will  be  equal  to  this  AB.  As- 
sume  in  AC,  and  BD  produced  two 
sects  CR,  DX,  equal  to  these  AC,  BD ; 
and  join  RX.  We  may  easily  show 
that  the  join  RX  will  be  equal  to  this 
AB, 


L- 


,k:,,_. 


and  the  angles  at  it  right. 


Fig.  4. 
And  first  indeed  by 


29 


superpositionem  quadrilateri  ABDC  super  quadrilaterum 
CDXR,  adhibita  communi  basi  CD.  Deinde  elegantius 
sic  proceditur.  Jungantur  AD,  RD.  Constat  (ex  quarta 
primi)  aequales  fore  in  triangulis  ACD,  RCD,  bases  AD, 
RD,  atque  item  angulos  CDA,  CDR,  ac  propterea  aequa- 
les  reliquos  ad  unum  rectum,  nimirum  ADB,  RDX. 
Quare  rursum  (ex  eadem  quarta  primi)  aequalis  erit,  in 
triangulis  ADB,  RDX,  basis  AB,  basi  RX.  Igitur  (ex 
praecedente)  anguli  ad  junctam  RX  erunt  recti,  ac  prop- 
terea  persistemus  in  eadem  hypothesi  anguH  recti. 

Quoniam  vero  augeri  simiHter  potest  longitudo  per- 
pendiculorum  in  infinitum,  sub  eadem  basi  AB,  consis- 
tente  semper  hypothesi  anguH  recti,  demonstrandum  est 
eandem  hypothesim  semper  mansuram  in  casu  cujusvis 
imminutionis  eorundem  perpendiculorum ;  quod  quidem 
ita  evincitur.  [7] 

Sumantur  in  AR,  et  BX  duo  quaeHbet  aequaHa  per- 
pendicula  AL,  BK,  jungaturque  LK.  Si  anguH  ad  junc- 
tam  LK  recti  non  sint,  erunt  tamen  (ex  prima  hujus)  in- 
vicem  aequales.  Erunt  igitur  ex  una  parte,  ut  puta  ver- 
sus  AB  obtusi,  et  versus  RX  acuti,  ut  nimirum  anguH 
hinc  inde  ad  utrunque  iHorum  punctorum  aequales  sint 
(ex  decimatertia  primi)  duobus  rectis.  Constat  autem 
aequaHa  etiam  invicem  esse  perpendicula  LR,  KX,  ipsi 
RX  insistentia.  Igitur  (ex  tertia  hujus)  erit  LK  major 
quidem  contraposita  RX,  et  minor  contraposita  AB. 

Hoc  autem  absurdum  est;  cum  AB,  et  RX  ostensae 
sint  aequales.  Non  ergo  mutabitur  hypothesis  anguH 
recti  sub  quacunque  imminutione  perpendiculorum,  dum 
consistat  semel  posita  basis  AB. 

Sed  neque  immutabitur  hypothesis  anguH  recti,  sub 
quacunque  imminutione,  aut  majori  ampHtudine  basis; 
cum  manifestum  sit  considerari  posse  ut  basim  quodvis 


superposition  of  the  quadrilateral  ABDC  upon  the  quad- 
rilateral  CDXR,  applied  to  the  common  base  CD. 

Also  we  may  proceed  more  elegantly  thus.  Join 
AD,  RD.  It  follows  (Eu.  I.  4)  in  the  triangles  ACD, 
RCD,  the  bases  AD,  RD  will  be  equal  and  Hkewise  the 
angles  CDA,  CDR,  and  certainly  ADB,  RDX  because 
equal  remainders  from  a  right  angle.  Whereby  in  turn 
(Eu.  I.  4)  in  the  triangles  ADB,  RDX,  the  base  AB  will 
be  equal  to  the  base  RX.  Therefore  (P.  IV.)  the  angles 
at  the  join  RX  will  be  right,  and  so  we  abide  in  the  same 
hypothesis  of  right  angle. 

Since  now  the  length  of  the  perpendiculars  can  be 
similarly  increased  infinitely,  under  the  same  base  AB, 
the  hypothesis  of  right  angle  always  subsisting,  it  only 
remains  to  be  proved  that  the  same  hypothesis  will  always 
abide  in  any  case  of  diminution  of  those  perpendiculars ; 
which  indeed  is  thus  evinced.  [7] 

Assume  in  AR,  and  BX  any  two  equal  perpendiculars 
AL,  BK,  and  join  LK.  If  the  angles  at  the  join  LK 
are  not  right,  nevertheless  (P.  I.)  they  will  be  equal  to 
each  other.  Therefore  they  will  be  toward  one  part,  as 
suppose  toward  AB  obtuse,  and  toward  RX  acute,  since 
certainly  the  angles  here  at  each  of  those  points  are 
(Eu.  I.  13)  equal  to  two  rights. 

But  it  also  holds  that  the  perpendiculars  LR,  KX, 
those  standing  upon  RX,  will  be  mutually  equal.  There- 
fore  (P.  III.)  LK  will  be  greater  indeed  than  the  oppo- 
site  RX,  and  less  than  the  opposite  AB. 

But  this  is  absurd;  because  AB,  and  RX  have  been 
shown  equal.  Therefore  the  hypothesis  of  right  angle 
is  not  changed  by  any  diminution  of  the  perpendiculars, 
whilst  abides  the  once  posited  base  AB. 

But  neither  is  the  hypothesis  of  right  angle  changed 
for  any  diminution,  or  greater  ampHtude  of  the  base; 
since  manifestly  may  be  considered  as  base  any  perpen- 

31 


perpendiculum  BK,  aut  BX,  atque  ideo  considerari  vi- 
cissim  ut  perpendicula  ipsam  AB,  et  rectam  aequalem 
contrapositam  KL,  aut  XR. 

Constat  igitur  hypothesim  anguH  recti,  si  vel  in  uno 
casu  sit  vera,  semper  in  omni  casu  illam  solam  esse  veram. 
Quod  erat  demonstrandum. 

PROPOSITIO  VI. 

Hypothesis  anguli  obHtsi,  si  vel  in  uno  casu  est  vera,  sem- 
per  in  omni  casu  illa  sola  est  vera. 

Demonstratur.  Efficiat  juncta  CD  (fig.  5.)  angulos 
obtusos  cum  duobus  quibusvis  aequaHbus  perpendicuHs 
AC,  BD,  uni  cuivis  rectae  AB  insistentibus.  Erit  CD 
(ex  tertia  hujus)  minor  ipsa  AB.  Sumantur  in  AC,  BD 
protractis  duae  quaeHbet  invicem  aequales  portiones  CR, 
[8]  DX ;  jungaturque  RX.  Jam  quaero  de  anguHs  ad  junc- 
tam  RX,  qui  utique  (ex  prima  hujus)  aequales  invicem 
erunt.  Si  obtusi  sunt,  habemus  intentum.  At  recti  non 
sunt;  quia  sic  unum  haberemus  casum  pro  hypothesi 
anguH  recti,  qui  nuHum  (ex  praecedente)  reHnqueret 
locum  pro  hypothesi  anguH  obtusi.  Sed  neque  acuti 
sunt.  Nam  sic  esset  RX  (ex  tertia  hujus)  major  ipsa 
AB ;  ac  propterea  multo  major  ipsa  CD.  Hoc  autem  sub- 
sistere  non  posse  sic  ostenditur.  Si  quadrilaterum  CDXR 
inteHigatur  impleri  rectis  abscindentibus  ab  ipsis  CR,  DX, 
portiones  invicem  aequales,  impHcat  transiri  a  recta  CD, 
quae  minor  est  ipsa  AB,  ad  RX  eadem  majorem,  quin 


32 


dicular  BK,  or  BX,  and  tberefore  may  be  considered  in 
turn  as  perpendiculars  that  AB,  and  the  equal  opposite 
sect  KL,  or  XR. 

Therefore  is  estabHshed  that  if  even  in  a  single  case 
the  hypothesis  of  right  angle  be  true,  always  in  every 
case  it  alone  is  true. 

Quod  erat  demonstrandum. 

PROPOSITION  VI. 

//  even  in  a  single  case  the  hypothesis  of  ohtuse  angle 
is  true,  always  in  every  case  it  alone  is  true, 

Proof.    Let  the  join  CD  (fig.  5)  make  obtuse  angles 
with  any  two  equal  perpendiculars  AC, 
BD,  standing  upon  any  straight  AB. 

CD  will  be  (P.  IIL)  less  than  this 
AB. 

Assume  in  AC  and  BD  produced 
any  two  mutually  equal  portions  CR 
[8]  and  DX;and  joinRX. 

Now  I  investigate  the  angles  at  the 


join  RX,  which  certainly  (P.  L)  will  Fig.  5. 

be  mutually  equal. 

If  they  are  obtuse  we  have  our  assertion. 

But  they  are  not  right;  because  thus  we  would  have 
a  case  for  the  hypothesis  of  right  angle,  which  (P.  V.) 
would  leave  no  place  for  the  hypothesis  of  obtuse  angle. 
But  neither  are  they  acute. 

For  thus  RX  would  be  (P.  III.)  greater  than  this 
AB ;  and  still  more  therefore  greater  than  CD  itself.  But 
that  this  cannot  be  is  thus  shown.  If  the  quadrilateral 
CDXR  is  taken  to  be  filled  up  by  straights  cutting  off 
from  these  CR,  DX,  portions  mutually  equal,  this  implies 
transition  from  the  sect  CD,  which  is  less  than  AB  itself, 
to  RX  greater  than  it,  verily  transition  through  a  certain 

33 


transeatur  per  quandam  ST  ipsi  AB  aequalem.  Hoc 
autem  absurdum  esse  in  hac  hypothesi  ex  eo  constat ;  quia 
sic  (ex  quarta  hujus)  unus  haberetur  casus  pro  hypo- 
thesi  anguli  recti,  qui  nullum  (ex  praecedente)  reHn- 
queret  locum  hypothesi  anguH  obtusi.  Igitur  anguH  ad 
junctam  RX  debent  esse  obtusi. 

Deinde,  sumptis  in  AC,  BD,  aequaHbus  portionibus 
AL,  BK;  simiH  modo  ostendemus  angulos  ad  junctam 
LK  nequire  esse  acutos  versus  ipsam  AB;  quia  sic  iHa 
foret  major,  quam  AB,  ac  propterea  multo  major  recta 
CD.  Hinc  autem  reperiri  deberet,  ut  supra,  quaedam 
intermedia  inter  CD  minorem,  et  LK  majorem  ipsa  AB ; 
intermedia,  inquam,  aequaHs  ipsi  AB,  quae  utique,  ex  jam 
notis,  omnem  locum  auferret  hypothesi  anguH  obtusi. 
Tandem  propter  hanc  ipsam  causam  recti  esse  nequeunt 
anguH  ad  junctam  LK;  ergo  erunt  obtusi.  Igitur  sub 
eadem  basi  AB,  auctis,  aut  imminutis  ad  Hbitum  perpen- 
dicuHs,  manebit  semper  hypothesis  anguH  obtusi. 

Sed  debet  idem  demonstrari  sub  assumpta  quaHbet 
basi.  EHgatur  (fig.  6.)  pro  basi  quodHbet  ex  praedictis 
perpendicuHs,  ut  puta  BX.  Dividantur  bifariam  in  punc- 
tis  [9J  M,  et  H  ipsae  AB,  RX;  jungaturque  MH.  Erit 
MH  (ex  secunda  hujus)  perpendicularis  ipsis  AB,  RX. 
Est  autem  angulus  ad  punctum  B  rectus  ex  hypothesi ;  et 
obtusus,  ex  jam  demonstratis,  ad  punctum  X.  Fiat  igitur 
angulus  rectus  BXP  versus  partes  ipsius  MH.  Occurret 
XP  ipsi  MH  in  quodam  puncto  P  inter  puncta  M,  et  H 


ST  equal  to  this  AB.  But  that  this  is  absurd  in  the  pres- 
ent  hypothesis  follows  so;  because  thus  (P.  IV.)  we  have 
a  case  for  the  hypothesis  of  right  angle,  which  (P.  V.) 
would  leave  no  place  for  the  hypothesis  of  obtuse  angle, 
Therefore  the  angles  at  the  join  RX  must  be  obtuse. 

Then,  equal  portions  AL,  BK  being  assumed  in  AC, 
BD;  in  a  similar  manner  we  show  the  angles  at  the  join 
LK  cannot  be  acute  toward  this  AB;  because  thus  it 
would  be  greater  than  AB,  and  still  more  therefore 
greater  than  the  sect  CD.  But  here  would  be  found,  as 
above,  a  certain  intermediate  between  CD  less,  and  LK 
greater  than  this  AB ;  an  intermediate,  I  say,  equal  to  AB 
itself,  which  certainly,  from  what  was  just  now  observed, 
would  take  away  every  place  for  the  hypothesis  of  obtuse 
angle. 

Finally  from  this  very  cause  the  angles  at  the  join 
LK  cannot  be  right ;  theref ore  they  will  be  obtuse. 

Therefore  with  the  same  base  AB,  the  perpendiculars 
being  increased  or  diminished  at  will,  the  hypothesis  of 
obtuse  angle  will  always  persist. 

But  the  same  ought  to  be  demonstrated  for  any  as- 
sumed  base. 

Let  there  be  chosen  (fig.  6)  for 
base  any  one  of  the  aforesaid  perpen- 
diculars,  as  BX  suppose. 

Let  AB,  RX  be  bisected  in  the 
points  [9]  M  and  H ;  and  MH  joined. 
MH  will  be  (P.  11.)  perpendicular  to 
AB,  RX.  But  the  angle  at  the  point 
B  is  right  by  hypothesis;  and  at  the 
point  X  obtuse,  from  what  has  just  now  been  demon- 
strated. 

Make  therefore  the  right  angle  BXP  toward  the  parts 
of  this  MH.  XP  will  meet  MH  itself  in  some  point  P 
situated  between  the  points  M  and  H;  since  on  the  one 


35 


constituto;  cum  ex  una  parte  angulus  BXH  sit  obtusus; 
et  ex  altera,  si  jungatur  XM,  angulus  BXM  (ex  decima- 
septima  primi)  sit  acutus.  Tum  vero;  quoniam  quadri- 
laterum  XBMP  tres  continet  angulos  rectos  ex  jam  notis, 
et  unum  obtusum  (ex  decimasexta  primi)  in  puncto  P, 
quia  est  externus  relate  ad  internum,  et  oppositum  rec- 
tum  angulum  in  puncto  H  trianguli  PHX ;  erit  latus  XP 
(ex  Cor.  I.  post  tertiam  hujus)  minus  contraposito  BM. 
Quare;  assumpta  in  BM  portione  BF  aequali  ipsi  XP; 
erunt  (ex  prima  hujus)  anguli  ad  junctam  PF  invicem 
aequales,  nimirum  obtusi,  cum  angulus  BFP  (ex  decima- 
sexta  primi)  sit  obtusus  propter  rectum  angulum  inter- 
num,  et  oppositum  FMP.  Igitur  sub  qualibet  basi  BX 
consistit  hypothesis  anguli  obtusi. 

Consistet  autem,  ut  supra,  eadem  hypthesis  sub  eadem 
basi  BX,  quamvis  aequalia  perpendicula  ad  libitum  auge- 
antur,  aut  minuantur.  Itaque  constat  hypothesim  anguli 
obtusi,  si  vel  in  uno  casu  sit  vera,  semper  in  omni  casu 
illam  solam  esse  veram.    Quod  erat  demonstrandum. 

PROPOSITIO  VII. 

Hypothesis  anguli  acuti,  si  vel  in  uno  casu  est  vera,  sem- 
per  in  omni  casu  illa  sola  est  vera. 

Demonstratur  facillime.  Si  enim  hypothesis  anguli 
acuti  permittat  aliquem  casum  alterutrius  hypothesis  aut 
anguli  recti,  aut  anguli  obtusi,  jam  (ex  duabus  praeceden- 
ClOltibus)  nullus  relinquetur  locus  ipsi  hypothesi  anguli 
acuti ;  quod  est  absurdum.  Itaque  hypothesis  anguli  acuti, 
si  vel  in  uno  casu  est  vera,  semper  in  omni  casu  illa  sola 
cst  vera.    Quod  erat  demonstrandum. 


hand  the  angle  BXH  is  obtuse ;  and,  on  the  other,  if  XM 
be  joined,  the  angle  BXM  (Eu.  I.  17)  is  acute.  Then 
however,  since  the  quadrilateral  XBMP  contains  three 
right  angles,  from  what  has  just  now  been  noted,  and 
one  obtuse  (Eu.  I.  16)  at  the  point  P,  because  it  is  exter- 
nal  in  relation  to  the  internal  and  opposite  right  angle  at 
the  point  H  of  the  triangle  PHX;  the  side  XP  will  be 
(Cor.  I.,  P.  ni.)  less  than  the  opposite  BM.  Wherefore, 
assuming  in  BM  the  portion  BF  equal  to  this  XP,  the 
angles  at  the  join  PF  will  be  (P.  I.)  mutually  equal, 
certainly  obtuse,  since  the  angle  BFP  (Eu.  I.  16)  is  ob- 
tuse  because  of  the  right  angle  interior  and  opposite 
FMP.  Therefore  the  hypothesis  of  obtuse  angle  abides 
for  any  base  BX. 

But,  as  above,  this  hypothesis  abides  for  this  base 
BX,  however  much  the  equal  perpendiculars  are  aug- 
mented  or  diminished  at  will.  Therefore  it  holds,  that 
if  even  in  a  single  case  the  hypothesis  of  obtuse  angle 
is  true,  always  in  every  case  it  alone  is  true. 

Quod  erat  demonstrandum. 

PROPOSITION  VII, 

//  even  in  a  single  case  the  hypothesis  of  acute  angle 
is  true,  always  in  every  case  it  alone  is  true. 

Proof  is  very  easily  given.  For  if  the  hypothesis  of 
acute  angle  should  permit  any  case  of  either  other  hypoth- 
esis,  either  of  right  angle,  or  of  obtuse  angle,  now  (from 
the  two  preceding  [10]  propositions)  no  place  would  be  left 
f or  the  hypothesis  of  acute  angle ;  which  is  absurd. 

Therefore  if  even  in  a  single  case  the  hypothesis  of 
acute  angle  is  true,  always  in  every  case  it  alone  is  true. 

Quod  erat  demonstrandum. 


37 


PROPOSITIO  VIII. 

Dato  quovis  triangulo  (fig.  7.)  ABD,  rectangulo  in  B, 
protrahatur  DA  usque  ad  aliquod  punctum  X,  et  per 
A  erigatur  ipsi  AB  perpendicularis  HAC,  existente 
puncto  H  ad  partes  anguli  XAB.  Dico  angulum  ex- 
ternum  XAH  aequalem  fore,  aut  minorem,  aut  ma- 
jorem  interno,  et  opposito  ADB,  prout  vera  sit  hypo- 
thesis  anguli  recti,  aut  anguli  obtusi,  aut  anguli  acuti : 
Et  vicissim. 

Demonstratur.  Sumatur  in  HC  portio  AC  aequalis 
ipsi  BD,  jungaturque  CD.  Erit  CD,  in  hypothesi  anguli 
recti,  aequaHs  (ex  tertia  hujus)  ipsi  AB.  Quare  angulus 
ADB  aequaHs  erit  (ex  octava  primi)  angulo  DAC,  sive 
ejus  aequali  (ex  decimaquinta  primi)  angulo  XAH. 
Quod  erat  primo  loco  demonstrandum. 

Tum,  in  hypothesi  anguH  obtusi,  erit  CD  (ex  eadem 
tertia  hujus)  minor  ipsa  AB.  Quare  in  trianguHs  ADB, 
DAC  erit  (ex  vigesimaquinta  primi)  angulus  DAC,  sive 
(ipsi  ad  verticem)  XAH,  minor  angulo  ADB.  Quod 
erat  secundo  loco  demonstrandum. 

Tandem,  in  hypothesi  anguH  acuti,  erit  CD  (ex  eadem 
tertia  hujus)  major  contraposita  AB.  Quare  in  prae- 
dictis  trianguHs,  erit  (ex  eadem  vigesimaquinta  primi) 
angulus  DAC,  sive  (ipsi  ad  verticem)  XAH,  major  an- 
gulo  ADB.    Quod  erat  tertio  loco  demonstrandum. 

Vicissim  autem :  si  angulus  CAD,  sive  ejus  ad  verti- 
cem  XAH,  aequaHs  sit  interno,  et  opposito  ADB;  erit 
(ex  quarta  primi)  juncta  CD  aequaHs  ipsi  AB,  ac  propte- 
[11]  rea  (ex  quarta  hujus)  vera  erit  hypothesis  anguH 
recti. 


PROPOSITION  VIII. 

Given  any  triangle  (fig.  7)  ABD,  right-angled  at  B;  pro- 
long  DA  to  any  point  X,  and  through  A  erect  HAC 
perpendicular  to  AB,  the  point 
H  being  within  the  angle  XAB. 
I  say  the  external  angle  XAH 
will  be  equal  to,  or  less,  or 
greater  than  the  internal  and  op- 
posite  ADB,  according  as  is  true 
the  hypothesis  of  right  angle,  or 
obtuse  angle,  or  acute  angle :  and 
inversely. 

Proof.  Assume  in  HC  the  portion  AC  equal  to  BD, 
and  join  CD.  CD  will  be,  in  the  hypothesis  of  right 
angle  (P.  III.)  equal  to  AB.  Wherefore  the  angle  ADB 
will  be  equal  (Eu.  I.  8)  to  the  angle  DAC,  or  to  its  equal 
(Eu.  I.  15)  the  angle  XAH. 

Quod  erat  primo  loco  demonstrandum. 

Then,  in  the  hypothesis  of  obtuse  angle,  CD  will  be 
(P.  III.)  less  than  AB. 

Wherefore  in  the  triangles  ADB,  DAC  the  angle 
DAC,  or  its  vertical  XAH,  will  be  (Eu.  I.  25)  less  than 
the  angle  ADB. 

Quod  erat  secundo  loco  demonstrandum. 

Finally,  in  the  hypothesis  of  acute  angle,  CD  will  be 
(P.  III.)  greater  than  the  opposite  AB.  Wherefore  in 
the  said  triangle  the  angle  DAC,  or  its  vertical  XAH,  will 
be  (Eu.  I.  25)  greater  than  the  angle  ADB. 

Quod  erat  tertio  loco  demonstrandum. 

But  inversely :  if  the  angle  CAD,  or  its  vertical  XAH, 
be  equal  to  the  internal  and  opposite  ADB;  the  join  CD 
will  be  (Eu.  I.  4)  equal  to  AB,  and  therefore  [11]  the  hy- 
pothesis  of  right  angle  will  be  (P.  IV.)  true. 

39  ... 


Sin  vero  angulus  CAD,  sive  ejus  ad  verticem  XAH, 
minor  sit,  aut  major  interno,  et  opposito  ADB ;  erit  etiam 
(ex  vigesimaquarta  primi)  juncta  CD  minor,  aut  major 
ipsa  AB;  ac  propterea  (ex  quarta  hujus)  vera  erit  re- 
spective  hypothesis  aut  anguli  obtusi,  aut  anguH  acuti 
Quae  omnia  erant  demonstranda. 

PROPOSITIO  IX. 

Cujusvis  trianguli  rectanguli  reliqui  duo  acuti  anguli  si- 
mul  sumpti  aequales  sunt  uni  recto,  in  hypothesi 
anguli  recti;  majores  uno  recto,  in  hypothesi  anguli 
obtusi;  minores  autem  in  hypothesi  anguli  acuti. 

Demonstratur.  Si  enim  angulus  XAH  (manente  fi- 
gura  superioris  Propositionis)  aequaHs  est  (nimirum,  ex 
praecedente,  in  hypothesi  anguH  recti)  angulo  ADB;  jam 
angulus  ADB  duos  rectos  efficiet  cum  angulo  HAD,  prout 
eos  efficit  (ex  decimatertia  primi)  praedictus  angulus 
XAH  cum  eodem  angulo  HAD.  Quare,  dempto  recto 
angulo  HAB,  aequales  manebunt  uni  recto  duo  simul 
anguH  ADB,  et  BAD.     Quod  erat  primum. 

Tum  vero;  si  angulus  XAH  minor  est  (nimirum,  ex 
praecedente,  in  hypothesi  anguH  obtusi)  angulo  ADB, 
jam  angulus  ADB  plusquam  duos  rectos  efficiet  cum 
angulo  HAD,  cum  quo  duos  efficit  rectos  (ex  praedicta 
decimatertia  primi)  angulus  XAH.  Quare,  dempto  an- 
gulo  HAB,  majores  erunt  uno  recto  duo  simul  anguH 
ADB,  et  BAD.    Quod  erat  secundum. 

Tandem,  si  angulus  XAH  major  sit  (nimirum,  ex 
praecedente,  in  hypothesi  anguH  acuti)  angulo  ADB;  jam 
angulus  ADB  minus  quam  duos  rectos  efficiet  cum  angulo 
HAD,  cum  quo  duos  efficit  rectos   (ex  eadem  decima- 


But  if  however  the  angle  CAD,  or  its  vertical  XAH, 
be  less,  or  greater  than  the  internal  or  opposite  ADB; 
also  the  join  CD  will  be  (Eu.  I.  24)  less  or  greater  than 
AB;  and  therefore  (P.  IV.)  will  be  true  respectively  the 
hypothesis  of  obtuse  angle,  or  acute  angle. 

Quae  omnia  erant  demonstranda. 

PROPOSITION  IX. 

In  any  right-angled  triangle  the  two  acute  angles  re- 
maining  are,  taken  together,  equal  to  one  right  angle, 
in  the  hypothesis  of  right  angle;  greater  than  one 
right  angle,  in  the  hypothesis  of  obtuse  angle;  but 
less  in  the  hypothesis  of  acute  angle. 

Proof.  For  if  the  angle  XAH  (fig.  7)  is  equal  to 
the  angle  ADB,  which  is  certain  from  the  preceding 
proposition  in  the  hypothesis  of  right  angle,  then  the 
angle  ADB  makes  up  with  the  angle  HAD  two  right 
angles,  as  (Eu.  I.  13)  the  aforesaid  angle  XAH  makes 
them  up  with  this  angle  HAD.  Wherefore,  the  right 
angle  HAB  being  subtracted,  the  two  angles  ADB  and 
BAD  remain  together  equal  to  one  right  angle. 

Quod  erat  primum. 

However,  if  the  angle  XAH  is  less  than  the  angle 
ADB,  which  is  certain  from  the  preceding  proposition 
in  the  hypothesis  of  obtuse  angle,  then  the  angle  ADB 
makes  up  with  the  angle  HAD  more  than  two  right 
angles,  since  with  it  (Eu.  I.  13)  the  angle  XAH  makes 
up  two.  Wherefore,  the  angle  HAB  being  subtracted, 
the  two  angles  ADB  and  BAD  will  be  together  greater 
than  one  right  angle. 

Quod  erat  secundum. 

Finally,  if  the  angle  XAH  be  greater  than  the  angle 
ADB,  which  is  certain  from  the  preceding  proposition 
in  the  hypothesis  of  acute  angle,  then  the  angle  ADB  will 
make  up  less  than  two  right  angles  with  the  angle  HAD, 


[12]tertia  primi)  angulus  XAH.  Quare,  dempto  angulo 
recto  HAB,  minores  erunt  uno  recto  duo  simul  anguli 
ADB,  et  BAD.    Quod  erat  tertium. 

PROPOSITIO  X. 

Si  recta  DB  (fig.  8.)  perpendiculariter  insistat  cuidam 
ABM,  sitque  juncta  DM  major  juncta  DA,  etiam 
basis  BM  major  erit  basi  BA.    Et  vicissim. 

Demonstratur.  Et  primo  quidem  non  erunt  illae  bases 
invicem  aequales.  Caeterum  (ex  quarta  primi)  aequales 
forent,  contra  hypothesim,  ipsae  AD,  DM.  Sed  neque 
erit  BA  major  quam  BM.  Caeterum,  sumpta  in  BA  por- 
tione  BS  aequaH  ipsi  BM,  junctaque  SD,  aequales  forent 
(ex  eadem  quarta  primi)  anguli  BSD,  BMD:  Est  autem 
angulus  BSD  (ex  decimasexta  primi)  major  angulo 
BAD.  Ergo  eodem  major  foret  angulus  BMD.  Hoc 
autem  est  contra  decimamoctavam  primi;  cum  latus  DM 
in  triangulo  MDA  supponatur  majus  latere  DA.  Restat 
igitur,  ut  basis  BM  major  sit  basi  BA.  Quod  erat  primo 
loco  demonstrandum. 

Deinde  si  alterutra  basis,  ut  puta  BA  (ne  immutetur 
figura)  fingatur  major  altera  BM;  tunc  juncta  DS,  quae 
ex  BA  abscindat  portionem  SB  aequalem  ipsi  BM,  aequa- 
lis  erit  (ex  quarta  primi)  junctae  DM.  Rursum  obtusus 
erit  (ex  decimasexta  primi)  angulus  DSA,  et  acutus  (ex 
decimaseptima  ejusdem  primi)  anguhis  DAS.  Quare  (ex 


since  with  this  (Eu.  I.  13)  [12]  the  angle  XAH  makes 
up  two.  Wherefore,  subtracting  the  right  angle  HAB, 
the  angles  ADB  and  BAD  will  be  together  less  than  one 
right  angle. 

Quod  erat  tertium. 

PROPOSITION  X, 
//  the  straight  DB   (fig.  8)   stand  perpendicular  to  a 
straight  ABM,  and  the  join  DM  be  greater  than  the 
join  DA,  then  also  the  base  BM  will  be  greater  than 
the  base  BA.    And  inversely. 

Proof.  And  in  the  first  place  assuredly  these  bases 
will  not  be  mutually  equal.  Otherwise  (Eu.  I.  4)  AD 
and  DM  would  be  equal,  contrary  to  the  hypothesis. 


But  neither  will  BA  be  greater  than  BM.  Otherwise, 
in  BA  the  portion  BS  being  taken  equal  to  BM,  and  SD 
joined,  the  angles  BSD,  BMD  (Eu.  I.  4)  would  be  equal. 
But  angle  BSD  is  (Eu.  I.  16)  greater  than  angle  BAD. 
Therefore  angle  BMD  would  be  greater  than  angle  BAD. 
But  this  is  contrary  to  Eu.  I.  18 ;  since  side  DM  in  triangle 
MDA  is  supposed  greater  than  side  DA.  It  remains 
therefore,  that  the  base  BM  is  greater  than  the  base  BA. 

Quod  erat  primo  loco  demonstrandum. 

Next  if  either  base,  as  BA  suppose  (the  figure  need 
not  be  changed)  is  conceived  as  greater  than  the  other 
BM ;  then  the  join  DS,  which  cuts  off  f rom  BA  the  por- 
tion  SB  equal  to  BM,  will  be  equal  (Eu.  I.  4)  to  the 
join  DM.  Again  angle  DSA  will  be  obtuse  (Eu.  I.  16) 
and  angle  DAS  acute  (Eu.  I.  17). 

43 


decimanona  ejusdem)  erit  juncta  DA  major  juncta  DS, 
ejusque  supposita  aequali  juncta  DM.  Quod  erat  secundo 
loco  demonstrandum.    Itaque  constant  proposita.[13J 

PROPOSITIO  XI. 

Recta  AP  (quantaelibet  longitudinis)  secet  duas  rectas 
PL,  AD  (fig.  9.)  priorem  quidem  sub  recto  angulo 
in  P,  posteriorem  vero  in  A  sub  quovis  acuto  angulo 
convergente  ad  partes  ipsius  PL.  Dico  rectas  AD, 
PL  (in  hypothesi  anguli  recti)  in  aliquo  puncto,  et 
quidem  ad  finitam,  seu  terminatam  distantiam,  tan- 
dem  coituras,  si  protrahantur  versus  illas  partes,  ad 
quas  cum  subjecta  AP  duos  angulos  efficiunt  duobus 
rectis  minores. 

Demonstratur.  Protrahatur  DA  versus  alias  partes 
usque  ad  aliquod  punctum  X,  et  per  A  erigatur  ipsi  AP 
perpendicularis  HAC,  existente  puncto  H  ad  partes  an- 
guli  XAP.  Tum  in  AD  protracta  versus  partes  ipsius 
PL  sumantur  duo  aequalia  intervalla  AD,  DF,  demittan- 
turque  ad  subjectam  AP  perpendiculares  DB,  FM,  quae 
utique  cadent,  propter  decimamseptimam  primi,  ad  partes 


Wherefore  (Eu.  I.  19)  the  join  DA  will  be  greater 
than  the  join  DS,  and  the  join  supposed  equal  to  it  DM. 

Quod  erat  secundo  loco  demonstrandum.  Itaque  con- 
stant  proposita.  [1^1 

PROPOSITION  XI. 

Let  the  straight  AP  (as  long  as  you  choose)  cut  the  two 
straights  PLj  AD  (fig.  9),  the  first  indeed  at  right 
angles  in  P^  but  the  latter  at  A  in  any  acute  angle 
converging  toward  the  parts  PL.  I  say  the  straights 
AD,  PL  (in  the  hypothesis  of  right  angle)  will  at 
length  meet  in  some  point,  and  indeed  at  a  finite,  or 
terminated  distance,  if  they  are  prolonged  toward 
those  parts  on  which  they  make  with  the  transversal 
AP  two  angles  together  less  than  two  right  angles. 

Proof.  Prolong  DA  toward  the  other  parts  even  to 
some  point  X,  and  through  A  erect  to  AP  the  perpendic- 
ular  HAC,  the  point  H  being  toward  the  parts  of  the 
angle  XAP. 


Fig.  9. 


Then  in  AD  produced  toward  the  parts  of  PL  assume 
two  equal  intervals  AD,  DF,  and  let  fall  upon  the  trans- 
versal  AP  the  perpendiculars  DB,  FM,  which  certainly 
(Eu.  I.  17)  fall  toward  the  parts  of  the  acute  angle  DAP; 

45 


anguli  acuti  DAP;  jungaturque  DM.     Ostendere  debeo 
junctam  DM  aequalem  fore  ipsi  DF,  sive  DA. 

Et  primo  quidem  nequit  DM  major  esse  ipsa  DF. 
Caeterum  enim  angulus  DMF  minor  foret  (ex  decima- 
octava  primi)  angulo  DFM,  sive  ejus  aequali  (ex  octava 
hujus,  in  hypothesi  anguli  recti)  angulo  XAH,  sive  ejus 
ad  verticem  CAD.  Quare  (cum  anguli  CAM,  FMA 
ponantur  aequales,  utpote  recti)  reHquus  angulus  DMA 
major  foret  reliquo  angulo  DAM.  Hoc  autem  absurdum 
est  (contra  decimamoctavam  primi)  si  nempe  DM  pona- 
tur  major  ipsa  DF,  sive  DA. 

Sed  neque  erit  DM  minor  ipsa  DF.  Caeterum  angu- 
lus  DMF  major  foret  (ex  eadem  decimaoctava  primi) 
angulo  DFM,  sive  ejus  aequali  (ex  praedicta  octava 
hujus,  in  hypothesi  anguH  recti)  angulo  XAH,  sive  ejus 
ad  verticem  CAD.  Quare  rursum,  ut  supra,  reHquus 
angulus  [14]  DMA  non  major,  sed  minor  foret  reHquo 
angulo  DAM.  Hoc  autem  absurdum  est  (contra  eandem 
decimamoctavam  primi)  si  nempe  DM  ponatur  minor 
ipsa  DF,  sive  DA. 

Restat  igitur,  ut  juncta  DM  aequaHs  sit  ipsi  DF,  sive 
DA.  Quare  in  triangulo  DAM  aequales  erunt  (ex  quinta 
primi)  anguH  ad  puncta  A,  et  M;  atque  ideo  in  trian- 
guH's  DBA,  DBM,  rectanguHs  in  B,  aequales  erunt  (ex 
vigesimasexta  primi)  bases  AB,  BM.  Quod  quidem  hoc 
loco  intendebatur. 

Quoniam  igitur  (assumpto  in  AD  continuata  inter- 
vallo  AF  duplo  intervalli  AD)  perpendicularis  FM  ad 
subjectam  AP  demissa  abscindit  ex  AP  versus  P  basim 
AM  duplam  ilHus  AB,  quam  abscindit  perpendicularis 
demissa  ex  puncto  D;  manifestum  est  tot  vicibus  fieri 
posse  hanc  praecedentis  intervalli  duplicationem,  ut  sic  in 
ipsa  AD  continuata  deveniatur  ad  quoddam  punctum  T, 
ex  quo  perpendicularis  demissa  ad  continuatam  AP  ab- 
scindat  quandam  AR  majorem  ipsa  quantalibet  finita  AP. 

46 


and  join  DM.  I  should  show  that  the  join  DM  will  be 
equal  to  DF,  or  DA. 

And  in  the  first  place  indeed  DM  cannot  be  greater 
than  DF.  For  otherwise  the  angle  DMF  would  be  less 
(Eu.  I.  18)  than  the  angle  DFM,  or  its  equal  (P.  VIIL, 
in  the  hypothesis  of  right  angle)  the  angle  XAH,  or  its 
vertical  CAD.  Wherefore  (since  the  angles  CAM,  FMA 
are  assumed  equal,  as  being  right)  the  remaining  angle 
DMA  would  be  greater  than  the  remaining  angle  DAM. 
But  this  is  absurd  (against  Eu.  I.  18)  if  indeed  DM  is 
taken  greater  than  DF  or  DA. 

But  neither  will  DM  be  less  than  this  DF.  Otherwise 
the  angle  DMF  would  be  greater  (Eu.  I.  18)  than  the 
angle  DFM,  or  its  equal  (P.  VIII.,  in  hypothesis  of  right 
angle)  the  angle  XAH,  or  its  vertical  CAD.  Wherefore 
again,  as  above,  the  remaining  angle  [14]  DMA  will  not 
be  greater,  but  less  than  the  remaining  angle  DAM.  But 
this  is  absurd  (against  Eu.  I.  18)  if  indeed  DM  is  taken 
less  than  DF,  or  DA. 

It  remains  therefore,  that  the  join  DM  is  equal  to 
DF,  or  DA.  Wherefore  in  the  triangle  DAM  (Eu.  I.  5) 
the  angles  at  the  points  A,  and  M  will  be  equal ;  and  there- 
fore  in  the  triangles  DBA,  DBM,  right-angled  at  B,  the 
bases  AB,  BM  will  be  equal  (Eu.  I.  26).  This  indeed 
was  here  our  aim. 

Since  therefore  (assuming  in  AD  produced  the  inter- 
val  AF  double  the  interval  AD)  the  perpendicular  FM 
let  fall  on  the  transversal  AP  cuts  off  from  AP  toward 
P  a  base  AM  double  AB,  which  the  perpendicular  let  fall 
f rom  the  point  D  cuts  off ;  it  is  manif est  that  this  dupli- 
cation  of  the  preceding  interval  can  be  so  many  times 
repeated,  that  thus  in  AD  continued  we  attain  to  a  cer- 
tain  point  T,  from  which  the  perpendicular  let  fall  upon 
AP  prolonged  cuts  off  a  certain  AR  greater  than  the 
finite  AP  however  great. 

47 


Constat  autem  evenire  id  non  posse,  nisi  post  occursum 
ipsius  continuatae  AD  in  quoddam  punctum  L  ipsius  PL. 
Si  enim  punctum  T  consisteret  ante  illum  occursum,  de- 
beret  ipsa  perpendicularis  TR  secare  eandem  PL  in  quo- 
dam  puncto  K.  Tunc  autem  in  triangulo  KPR  inveniren- 
tur  duo  anguli  recti  in  punctis  P,  et  R ;  quod  est  absurdum 
contra  decimamseptimam  primi.  Itaque  constat  rectas 
AD,  PL  sibi  invicem  (in  hypothesi  anguli  recti)  in  aHquo 
puncto  occursuras  (et  quidem  ad  finitam,  seu  terminatam 
distantiam)  si  protrahantur  versus  illas  partes,  ad  quas 
cum  subjecta  AP  (quantaehbet  finitae  longitudinis)  duos 
angulos  efiiciunt  duobus  rectis  minores.  Quod  erat  de- 
monstrandum.  [15] 

PROPOSITIO  XII. 

Rursum  dico  rectam  AD  alicubi  ad  eas  partes  occursuram 
rectae  PL  (et  quidem  ad  finitam,  seu  terminatam  di- 
stantiam)  etiam  in  hypothesi  anguli  obtusi. 

Demonstratur.  Nam  sumpta,  ut  in  superiore  Propo- 
sitione,  DF  aequali  ipsi  AD,  demissisque  jam  notis  per- 
pendicularibus,  ostendere  debeo  junctam  DM  majorem 
fore  ipsa  DF,  sive  DA,  atque  ideo  (ex  decima  hujus)  rec- 
tam  BM  majorem  fore  ipsa  AB.  Et  primo  non  erit  DM 
aequalis  ipsi  DF.    Caeterum  angulus  DMF  aequalis  foret 


But  it  is  evident  this  cannot  happen,  except  after  the 
meeting  of  the  prolonged  AD  with  PL  in  some  point  L. 

For  if  the  point  T  occurred  before  that  meeting,  the 
perpendicular  TR  must  cut  PL  in  some  point  K.  But 
then  in  the  triangle  KPR  would  be  f  ound  two  right  angles 
at  the  points  P  and  R;  which  is  absurd  (against  Eu.  I. 
17). 

Therefore  it  holds  that  the  straights  AD,  PL  meet 
each  other  mutually  (in  the  hypothesis  of  right  angle) 
in  some  point  (and  indeed  at  a  finite  or  terminated  dis- 
tance)  if  they  be  produced  toward  that  side,  on  which 
with  the  transversal  AP  (of  finite  length  as  great  as 
you  choose)  they  make  two  angles  together  less  than  two 
right  angles. 

Quod  erat  demonstrandum.  [15j 


PROPOSITION  XII. 

Again  I  say  also  in  the  hypothesis  of  obtuse  angle  the 
straight  AD  will  meet  the  straight  PL  somewhere 
toward  those  parts  (and  indeed  at  a  finite,  or  termi- 
nated  distance) . 


Proof.  For,  as  in  the  preceding  proposition,  DF 
being  assumed  equal  to  AD,  and  the  just  noted  perpen- 
diculars  let  fall,  I  must  show  the  join  DM  will  be  greater 
than  DF,  or  DA,  and  therefore  (P.  X.)  the  straight  BM 
will  be  greater  than  AB. 

And  in  the  first  place  DM  will  not  be  equal  to  DF. 
Otherwise  the  angle  DMF  would  be  equal  (Eu.  I.  5)  to 

49 


(ex  quinta  primi)  angulo  DFM,  atque  ideo  major  (ex 
octava  hujus  in  hypothesi  anguli  obtusi)  angulo  externo 
XAH,  sive  ejus  ad  verticem  CAF.  Quare  (cum  anguli 
CAM,  FMA  ponantur  aequales  utpote  recti)  reliquus  an- 
gulus  DMA  minor  foret  reHquo  angulo  DAM.  Quod  est 
absurdum  contra  quintam  primi,  si  nempe  DM  aequalis 
sit  ipsi  DF,  sive  DA. 

Sed  neque  ipsa  DM  minor  est  altera  DF,  sive  DA. 
Caeterum  (ex  decimaoctava  primi)  angulus  DMF  major 
foret  angulo  DFM,  atque  ideo  (in  hac  hypothesi  anguli 
obtusi)  multo  major  angulo  externo  XAH,  sive  ejus  ad 
verticem  CAD.  Quare  rursum,  ut  supra,  reliquus  angu- 
lus  DMA  multo  minor  foret  reliquo  angulo  DAM.  Hoc 
autem  absurdum  est,  contra  eandem  decimamoctavam 
primi,  si  nempe  DM  minor  sit  ipsa  DF,  sive  DA. 

Restat  igitur,  ut  juncta  DM  major  sit  ipsa  DF,  sive 
DA,  atque  ideo  (ex  decima  hujus)  ipsa  BM  major  sit 
altera  AB.    Quod  erat  hoc  loco  intentum. 

Quoniam  igitur,  assumpto  in  AD  continuata  inter- 
vallo  AF  duplo  intervalli  AD,  perpendicularis  FM  ad 
subjectam  AP  demissa  plus  duplo  ex  eadem  abscindit, 
quam  abscindatur  a  perpendiculari  demissa  ex  puncto  D ; 
[16]  multo  citius  in  hac  hypothesi  anguli  obtusi,  quam  in 
superiore  hypothesi  anguli  recti,  devenietur  ad  tantum 
intervallum,  ex  quo  perpendicularis  demissa  abscindat 
basim  majorem  ipsa  quantalibet  designata  AP.  Hoc 
autem,  ut  in  superiore  Propositione,  contingere  nequit, 
nisi  post  occursum  continuatae  AD  in  aliquod  punctum 
ipsius  PL;  et  quidem  ad  finitam,  seu  terminatam  distan- 
tiam.    Quod  erat  etc. 


so 


the  angle  DFM,  and  therefore  greater  (P.  VIII. ,  in  the 
hypothesis  of  obtuse  angle)  than  the  external  angle  XAH, 
or  its  vertical  CAF. 

Wherefore  (since  the  angles  CAM,  FMA  are  taken 
equal,  as  being  right)  the  remaining  angle  DMA  would 
be  less  than  the  remaining  angle  DAM.  This  is  absurd 
(against  Eu.  I.  5),  if  indeed  DM  be  equal  to  DF,  or  DA. 

But  neither  is  DM  less  than  DF,  or  DA.  Otherwise 
(Eu.  I.  18)  the  angle  DMF  would  be  greater  than  the 
angle  DFM,  and  therefore  still  greater  (in  the  hypothesis 
of  obtuse  angle)  than  the  external  angle  XAH,  or  its 
vertical  CAD.  Wherefore  again,  as  above,  the  remain- 
ing  angle  DMA  would  be  still  less  than  the  remaining 
angle  DAM.  But  this  is  absurd  (against  Eu.  I.  18)  if 
indeed  DM  be  less  than  DF,  or  DA. 

It  remains  therefore,  that  the  join  DM  is  greater  than 
DF,  or  DA,  and  therefore  (P.  X.)  BM  is  greater  than 
AB. 

Ouod  erat  hoc  loco  intentum. 

Since  therefore,  assuming  in  AD  produced  the  inter- 
val  AF  double  the  interval  AD,  the  perpendicular  FM  let 
fall  on  the  transversal  AP  cuts  off  from  it  more  than 
double  what  is  cut  off  by  the  perpendicular  let  fall  from 
the  point  D:  [^6]  more  quickly  by  far  in  this  hypothesis 
of  obtuse  angle,  than  in  the  preceding  hypothesis  of  right 
angle,  we  attain  to  an  interval  so  great,  that  from  it  the 
perpendicular  let  fall  cuts  off  a  base  greater  than  the 
designated  AP  however  great. 

But  this,  as  in  the  preceding  proposition,  could  not 
happen,  unless  after  the  meeting  of  the  produced  AD 
with  PL  in  some  point;  and  indeed  at  a  finite,  or  termi- 
nated  distance. 

Quod  erat  etc. 


5t 


PROPOSITIO  XIII. 

Si  recta  XA  (quantaelibet  designatae  longitudinis)  inci- 
dens  in  duas  rectas  AD,  XL,  efficiat  cum  eisdem  ad 
easdem  partes  (fig.  11.)  angulos  internos  XAD, 
AXL  minores  duobus  rectis:  dico,  illas  duas  (etiamsi 
neuter  illorum  angulorum  sit  rectus)  tandem  in 
aliquo  puncto  ad  partes  illorum  angulorum  invicem 
coituras,  et  quidem  ad  finitam,  seu  terminatam  dis- 
tantiam,  dum  consistat  alterutra  hypothesis  aut  an- 
guli  recti,  aut  anguli  obtusi. 

Demonstratur.  Nam  unus  praedictorum  angulorum, 
ut  puta  AXL,  erit  acutus.  Itaque  ex  apice  alterius  an- 
guli  demittatur  ad  XL  perpendicularis  AP,  quae  utique 
(propter  decimamseptimam  primi)  cadet  ad  partes  an- 
guli  acuti  AXL.  Quoniam  igitur  in  triangulo  APX,  rect- 
angulo  in  P,  duo  simul  anguli  acuti  PAX,  PXA,  minores 
non  sunt  (ex  nona  hujus)  uno  recto,  in  utraque  hypo- 
thesi  aut  anguli  recti,  aut  anguli  obtusi ;  si  duo  isti  anguli 
auferantur  in  summa  angulorum  propositorum  jam  re- 
liquus  angulus  PAD  minor  erit  recto.  Itaque  erimus  in 
casu  duarum  praecedentium  Propositionum,  dum  scili- 
cet  alterutra  hypothesis  consistat  aut  anguli  recti,  aut 
anguli  obtusi.  Quare  (ex  eisdem)  rectae  AD,  et  PL, 
sive  XL,  in  aliquo  puncto  finitae,  seu  terminatae  distan- 
tiae  ad  notas  [17]  partes  concurrent,  tam  sub  una,  quam 
sub  altera  praedictarum  hypothesium.  Quod  erat  demon- 
strandum. 


PROPOSITION  XIII. 
//  the  straight  XA  (of  designated  length  however  great) 
meeting  two  straights  AD,  XL,  makes  with  them 
toward  the  same  parts  (fig.  II)  internal  anglesXAD, 
AXL  less  than  two  right  angles:  I  say,  these  two 
(even  if  neither  of  those  angles  be  a  right  angle) 
at  length  will  mutually  meet  in  some  point  on  the 
side  toward  those  angles,  and  indeed  at  a  finite,  or 
terminated  distance,  if  either  hypothesis  holds,  of 
right  angle  or  of  obtuse  angle. 
Proof.    For  one  of  the  said  angles,  as  AXL  suppose, 
will  be  acute. 

Accordingly  from  the  vertex  of  the  other  angle  is 
dropped  the  perpendicular  AP  on  XL,  which  certainly 


(because  of  Eu.  I.  17)  falls  on  the  side  of  the  acute  angle 
AXL.  Since  therefore  in  the  triangle  APX,  right-angled 
at  P,  the  two  acute  angles  PAX,  PXA,  together  are  not 
less  (P.  IX.)  than  a  right  angle,  in  either  hypothesis,  of 
right  angle,  or  of  obtuse  angle;  if  these  two  angles  are 
taken  away  from  the  sum  of  the  given  angles  the  then 
remaining  angle  PAD  will  be  less  than  a  right  angle. 
Consequently  we  will  be  in  the  case  of  the  two  preceding 
propositions,  since  it  is  obvious  that  one  or  the  other 
hypothesis  holds,  either  of  right  angle,  or  of  obtuse  angle. 

Wherefore  the  straights  AD,  and  PL,  or  XL,  meet 
in  some  point  at  a  finite,  or  terminated  distance  on  the 
side  noted,  [17]  as  well  under  the  one  as  under  the  other 
mentioned  hypothesis. 

Quod  erat  demonstrandum. 

53 


SCHOLION  1. 

Ubi  observare  licet  notabile  discrimen  ab  hypothesi 
anguli  acuti.  Nam  in  ista  demonstrari  nequiret  generalis 
hujusmodi  rectarum  concursus,  quoties  recta  aliqua  in 
duas  incidens,  duos  ad  easdem  partes  efficiat  internos 
angulos  duobus  rectis  minores;  nequiret,  inquam,  directe 
demonstrari,  etiamsi  in  eadem  hypothesi  admitteretur 
praedictus  generaHs  concursus,  quoties  unus  duorum  an- 
gulorum  est  rectus.  Quamvis  enim  recta  AD  perpen- 
dicularis  et  ipsa  f oret  rectae  AP ;  quo  casu  nequiret  certe, 
propter  17.  primi,  concurrere  cum  altera  perpendiculari 
PL;  nihilominus  duo  simul  anguH  DAX,  PXA,  minores 
forent  duobus  rectis,  juxta  hypothesim  praedictam,  cum 
in  ea  duo  simul  anguH  PAX,  PXA  minores  sint  (ex  nona 
hujus)  uno  recto.  Id  autem  observasse  operae  pretium 
fuit. 

QuaHter  vero  ex  eo  solo  admisso  generaH  concursu, 
dum  unus  angulorum  est  rectus,  et  quidem  sub  assignata 
quantumHbet  parva  incidente,  destrui  possit  hypothesis 
anguH  acuti ;  docebimus  post  tres  sequentes  Propositiones. 

SCHOLION  n. 

In  tribus  ante  jactis  theorematis  studiose  apposui 
illam  conditionem,  quod  recta  incidens  AP,  sive  XA, 
intehigatur  esse  quantaelibet  designatae  longitudinis.  Si 
enim,  citra  omnem  rectae  incidentis  determinatam  men- 
suram,  praecise  agatur  de  exhibendo,  ac  demonstrando 
duarum  rectarum  concursu  in  apicem  cujusdam  trianguH, 
cujus  [IS]  anguH  ad  basim  sint  dati  (minores  utique  duo- 
bus  rectis)  ut  puta  unus  rectus,  et  alter  duobus  tantum 


S4 


SCHOLION  I. 

Here  may  be  observed  a  notable  difference  from  the 
hypothesis  of  acute  angle. 

For  in  this  the  general  concurrence  of  such  straights 
cannot  be  demonstrated,  as  often  as  any  straight  falling 
upon  two,  makes  two  internal  angles  toward  the  same 
parts  less  than  two  right  angles ;  cannot,  I  say,  be  directly 
demonstrated,  even  if  in  this  hypothesis  the  aforesaid 
general  concurrence  be  admitted,  as  often  as  one  of  the 
two  angles  is  right. 

For  although  the  straight  AD  be  perpendicular  even 
to  the  straight  AP;  in  which  case  it  certainly  could  not 
concur  with  another  perpendicular  PL  (Eu.  I.  17) ; 
nevertheless  the  two  angles  together  DAX,  PXA,  could 
be  less  than  two  right  angles,  in  accordance  with  the 
aforesaid  hypothesis,  since  in  it  the  two  angles  together 
PAX,  PXA  may  be  less  (P.  IX.)  than  one  right  angle. 
But  it  was  worth  while  to  have  observed  this. 

But  how,  solely  from  the  general  admission  of  con- 
currence  when  one  of  the  angles  is  right,  and  with  an 
assigned  incident  however  small,  the  hypothesis  of  acute 
angle  can  be  demoHshed;  this  we  shall  show  after  the 
next  three  propositions. 

SCHOLION  II. 

In  the  three  preceding  theorems  I  have  studiously  set 
down  this  condition,  that  the  cutting  straight  AP,  or  XA, 
is  understood  to  be  of  a  designated  length  as  great  as  you 
choose. 

For  if,  without  any  determinate  extent  of  the  cutting 
straight,  it  be  discussed  precisely  concerning  the  exhibit- 
ing  and  demonstrating  of  the  concurrence  of  two  straights 
at  the  apex  of  a  certain  triangle,  whose  [^8]  angles  at  the 
hase  are  given  (less  indeed  than  two  right  angles)  as, 

55 


gradibus,  vel,  ut  libet,  minus  deficiens  a  recto;  quis  est 
tam  expers  Geometriae,  quin  statim  rem  ipsam  demon- 
strative  exhibeat?  Nam  supponatur  (fig.  12.)  datus  qui- 
libet  angulus  BAP,  ut  puta  88.  graduum.  Si  ergo  ex 
quolibet  puncto  B  ipsius  AB,  demittatur  ad  subjectam  AP 
(juxta  duodecimam  primi)  perpendicularis  BP,  constat 
enim  vero  in  eo  triangulo  ABP  exhibitum  fore  demon- 
strative  concursum  optatum  in  eo  puncto  B. 

Quod  si  alter  angulus  ad  basim  postuletur  et  ipse  mi- 
nor  recto,  ut  puta  84.  graduum,  quem  nempe  exhibeat 
datus  angulus  K:  tunc  (juxta  23.  primi)  efficere  poteris 
versus  partes  rectae  AB  aequalem  angulum  APD,  occur- 
rente  PD  ipsi  AB  in  quodam  ejus  intermedio  puncto  D. 
Quare  habebitur  rursum  demonstrative  concursus  optatus 
in  eo  puncto  D. 

Tandem  vero:  si  alter  angulus  postuletur  obtusus, 
sed  minor  tamen  92.  gradibus,  ne  cum  alio  dato  angulo 
B AP  compleantur  duo  recti :  exhibitus  hic  sit  in  quodam 
angulo  R  91.  graduum.  Ostendendum  est,  unum  ahquod 
esse  punctum  X  in  ipsa  AP,  ad  quod  juncta  BX  efficiat 
angulum  BXA  aequalem  dato  angulo  R  91.  graduum; 
adeo  ut  propterea  sub  quadam  recta  incidente  AX  habea- 
tur  concursus  optatus  in  praedicto  puncto  B.  Sic  autem 
proceditur.  Quandoquidem  (protracta  PA  usque  in  ali- 
quod  punctum  H)  angulus  externus  BAH  et  est  (propter 
decimamtertiam  primi)  92.  graduum,  cum  angulus  in- 
terior  BAP  positus  sit  88.  graduum;  ac  rursum,  propter 


56 


suppose,  one  right,  and  the  other  less  than  a  right  by  as 
much  as  two  degrees,  or,  if  you  please,  by  less;  who  is 
so  devoid  of  geometry  that  he  could  not  immediately 
show  the  thing  itself  demonstratively  ? 

For  suppose  (fig.  12)  given 
any  angle  BAP,  as,  say,  88  de- 
grees.  If  therefore  from  any 
point  B  of  this  AB,  is  let  fall  on 
the  base  AP  (Eu.  I.  12)  the 
perpendicular  BP,  it  holds  cer- 
tainly  that  in  this  triangle  ABP 
would  be  exhibited  demonstra- 
tively  the  desired  concurrence 
at  this  point  B. 

But  if  the  other  angle  at  the  base  is  postulated,  and 
is  less  than  a  right,  as,  suppose,  84  degrees,  which  indeed 
the  given  angle  K  represents:  then  (Eu".  I.  23)  one 
would  be  able  to  make  toward  the  parts  of  the  straight 
AB  an  equal  angle  APD,  PD  meeting  this  AB  in  D, 
some  intermediate  point  of  it.  Wherefore  the  desired 
concourse  is  again  obtained  demonstratively  in  this 
point  D. 

But  finally:  if  the  other  angle  is  postulated  obtuse, 
but  yet  less  than  92  degrees,  lest  with  the  other  given 
angle  BAP  it  should  make  up  two  rights:  this  may  be 
represented  in  a  certain  angle  R  of  91  degrees.  It  is  to 
be  shown,  that  there  is  some  point  X  of  this  AP,  to  which 
the  join  BX  makes  an  angle  BXA  equal  to  the  given 
angle  R  of  91  degrees :  so  that  therefore  under  a  certain 
cutting  straight  AX  the  desired  meeting  in  the  point  B 
may  be  obtained. 

Now  we  may  proceed  thus. 

PA  being  produced  to  any  point  H,  since  the  external 
angle  BAH  is  (Eu.  I.  13)  92  degrees,  because  the  interior 
angle  BAP  is  by  hypothesis  88  degrees;  and  again  (Eu. 

57 


decimamsextam  priml,  major  est  non  solum  angulo  recto 
BPA,  verum  etiam  quibusvis  eodem  titulo  obtusis  angulis 
BXA,  sumpto  puncto  X  ubilibet  intra  ipsam  PA,  et  qui- 
dem,  propter  eandem  decimamsextam  primi,  semper  ma- 
joribus,  dum  punctum  X  assumitur  propius  puncto  A: 
consequens  pla-[19]ne  est,  ut  inter  istos  angulos,  unum  90. 
graduum  in  puncto  P,  et  alterum  92.  graduum  in  puncto 
A,  unus  reperiatur  angulus  BXA,  qui  sit  91.  graduum, 
nimirum  aequalis  dato  angulo  R. 

Nihilominus,  omissa  postrema  hac  observatione  circa 
angulum  obtusum,  cavere  diligentissime  oportet,  in  eo 
positam  esse  difficultatem  illius  pronunciati  Euclidaei, 
quod  velit  occursum  duarum  rectarum;  in  illam  utique 
partem,  ad  quam  cum  recta  incidente  duos  angulos  effi- 
ciant  duobus  rectis  minores ;  atque  ita  quidem  praedictum 
occursum  veHt,  quantaecunque  longitudinis  sit  incidens 
assignata.  Caeterum  enim  (ut  jam  monui  in  praecedente 
Scholio)  demonstrabo  generalem  istum  occursum  ex  solo 
admisso  occursu  ejusmodi,  dum  unus  angulorum  sit  rec- 
tus ;  et  quidem,  etiamsi  admisso  non  pro  qualibet  assigna- 
bili  finita  incidente,  sed  solum  admisso  intra  limites  cujus- 
dam  assignatae  parvissimae  incidentis. 

PROPOSITIO  XIV. 

Hypothesis  angtdi  obtusi  est  ahsolute  falsa,  quia  se  ipsam 
destruit, 

Demonstratur.  Ex  hypothesi  anguli  obtusi,  assumpta 
ut  vera,  jam  elicuimus  veritatem  illius  Pronunciati  Eucli- 
daei;  quod  duae  rectae  sibi  invicem  in  aliquo  puncto  ad 
eas  partes  occursurae  sint,  ad  quas  recta  quaedam,  easdem 
secans,  duos  qualescunque  effecerit  internos  angulos,  duo- 
bus  rectis  minores.     Stante  autem  hoc  Pronunciato,  cui 


58 


I.  16)  is  greater  not  alone  than  the  right  angle  BPA  but 
also,  for  the  same  reason,  than  any  obtuse  angle  BXA, 
the  point  X  being  assumed  wherever  you  choose  within 
this  PA,  and  indeed  always  greater  as  the  point  X  is 
assumed  nearer  to  the  point  A  (Eu.  I.  16)  :  it  is  an 
evident  consequence,  [19]  that  between  those  angles,  one 
of  90  degrees  at  the  point  P,  and  the  other  of  92  degrees 
at  the  point  A,  one  angle  BXA  is  found,  which  is  91 
degrees,  truly  equal  to  the  given  angle  R. 

None  the  less,  omitting  here  the  last  observation  about 
the  obtuse  angle,  it  is  necessary  most  diligently  to  take 
care  that  the  difificulty  of  this  assumption  of  Euclid  be 
fixed  in  this,  that  it  asserts  the  mneting  of  two  straights ; 
in  particular  in  that  part  toward  which  they  make  with 
the  cutting  straight  two  angles  together  less  than  two 
right  angles;  and  assuredly  that  it  asserts  the  aforesaid 
meeting  thus,  of  whatever  length  be  the  assigned  trans- 
versal. 

However  (as  I  have  already  mentioned  in  the  pre- 
ceding  schoHon)  I  shall  demonstrate  the  general  meeting 
solely  from  the  admitted  meeting  of  this  sort  when  one 
of  the  angles  is  right;  and  indeed  even  if  it  be  admitted 
not  for  any  assignable  finite  transversal,  but  alone  ad- 
mitted  within  the  limits  of  any  assigned  very  small  trans- 
versal. 

PROPOSITION  XIV. 

The  hypothesis  of  obtuse  angle  is  absolutely  false,  be- 
cause  it  destroys  itself. 

Proof.  From  the  hypothesis  of  obtuse  angle,  as- 
sumed  as  true,  we  have  now  deduced  the  truth  of  Euclid's 
postulate:  that  two  straights  will  meet  each  other  in 
some  point  toward  those  parts,  toward  which  a  certain 
straight,  cutting  them,  makes  two  intemal  angles,  of 
whatever  kind,  less  than  two  right  angles. 

59 


innititur  Euclides  post  vigesimamoctavam  sui  Libri  primi, 
manifestum  est  omnibus  Geometris,  solam  hypothesim 
anguli  recti  esse  veram,  nec  ullum  relinqui  locum  hypo- 
thesi  anguH  obtusi.  Igitur  hypothesis  anguH  obtusi  est 
absolute  f  alsa,  quia  se  ipsam  destruit.  Quod  erat  demon- 
strandum.  [20] 

AHter,  ac  magis  immediate.  Quandoquidem  ex  hypo- 
thesi  anguH  obtusi  demonstravimus  (in  nona  hujus)  duos 
(fig.  11.)  acutos  angulos  trianguH  APX,  rectanguH  in  P, 
majores  esse  uno  recto;  constat  talem  assumi  posse  acu- 
tum  angulum  PAD,  qui  simul  cum  praedictis  duobus 
acutis  anguHs  duos  rectos  efficiat.  Tunc  autem  recta  AD 
deberet  (ex  praecedente,  juxta  hypothesim  anguH  obtusi) 
aHquando  concurrere  cum  ipsa  PL,  sive  XL,  respectu 
habito  ad  secantem,  sive  incidentem  AP;  quod  est  mani- 
festum  absurdum  contra  decimamseptimam  primi,  si  re- 
spicias  ad  secantem,  sive  incidentem  AX. 

PROPOSITIO  XV. 

Ex  quolibet  triangulo  ABC,  cujus  tres  simul  anguli  (fig, 
13.)  aequales  sint,  aut  majores,  aut  minores  duohus 
rectisj  stabilitur  respective  hypothesis  aut  anguli 
rectiy  aut  anguli  obtusi,  aut  anguli  acuti. 

Demonstratur.  Nam  duo  saltem  iUius  trianguH  an- 
guH,  ut  puta  ad  puncta  A,  et  C,  acuti  erunt,  propter  deci- 
mamseptimam  primi.  Quare  perpendicularis,  ex  apice 
reHqui  anguH  B  ad  ipsam  AC  demissa,  secabit  ipsam  AC 
(propter  eandem  decimamseptimam  primi)  in  aHquo 
puncto  intermedio  D.  Si  ergo  tres  anguH  ipsius  trianguH 
ABC  supponantur  aequales  duobus  rectis,  constat  aequales 


6o 


But  this  assumption  holding  good,  on  which  Euclid 
supports  himself  after  I.  28,  it  is  manifest  to  all  geometers 
that  the  hypothesis  of  right  angle  alone  is  true,  nor  any 
place  left  for  the  hypothesis  of  obtuse  angle.  Therefore 
the  hypothesis  of  obtuse  angle  is  absolutely  false,  because 
it  destroys  itself. 

Quod  erat  demonstrandum.  [20J 

Otherwise,  and  more  immediately.  Since  from  the 
hypothesis  of  obtuse  angle  we  have  proved  (P.  IX.)  that 
two  (fig.  11)  acute  angles  of  the  triangle  APX,  right- 
angled  at  P,  are  greater  than  one  right  angle ;  it  f ollows 
that  an  acute  angle  PAD  may  be  assumed  such,  that 
together  with  the  aforesaid  two  acute  angles  it  makes 
up  two  right  angles.  But  then  the  straight  AD  must  (by 
the  preceding  proposition,  joined  to  the  hypothesis  of 
obtuse  angle)  at  length  meet  with  this  PL,  or  XL,  regard 
being  had  to  the  secant,  or  incident  AP;  which  is  mani- 
festly  absurd  (against  Eu.  L  17)  if  we  regard  the  secant, 
or  incident  AX. 

PROPOSITION  XV. 

By  any  triangle  ABC,  of  which  the  three  angles  (fig.  13) 
are  equal  to,  or  greater,  or  less  than  two  right  an- 
gles,  is  estahlished  respectively 
the  hypothesis  of  right  angle,  yM 

or  obtuse  angle,  or  acute  angle. 

Proof.  For  anyhow  two  angles 
of  this  triangle,  as  suppose  at  the 
points  A  and  C,  will  be  acute  (Eu. 
I.  17).     Wherefore  the  perpendic-  Fig.  13. 

ular,  let  fall  from  the  apex  of  the 
remaining  angle  B  upon  AC,  will  cut  AC  (Eu.  I.  17) 
in  some  intermediate  point  D. 

If  therefore  the  three  angles  of  this  triangle  ABC 
are  supposed  equal  to  two  right  angles,  it  follows  that 

6t 


tj 


fore  quatuor  rectis  omnes  simul  angulos  triangulorum 
ADB,  CDB,  propter  duos  additos  rectos  angulos  ad 
punctum  D.  Hoc  stante :  neutrius  modo  dictorum  trian- 
gulorum,  ut  puta  ADB,  tres  simul  anguli  minores  erunt, 
aut  majores  duobus  rectis ;  nam  sic  viceversa  alterius  tri- 
anguli  tres  simul  anguli  majores  forent,  aut  minores  duo- 
bus  rectis.  Quare  (ex  nona  hujus)  ab  uno  quidem 
triangulo  stabiliretur  hypothesis  anguH  acuti,  et  ab  altero 
hypothesis  anguH  [21]  obtusi;  quod  repugnat  sextae,  et 
septimae  hujus.  Igitur  tres  simul  anguH  utriusque  prae- 
dictorum  triangulorum  aequales  erunt  duobus  rectis;  ac 
propterea  (ex  nona  hujus)  stabiHetur  hypothesis  anguH 
recti.    Quod  erat  primo  loco  demonstrandum. 

Sin  autem  tres  anguH  propositi  trianguH  ABC  ponan- 
tur  majores  duobus  rectis;  jam  duorum  triangulorum 
ADB,  CDB  omnes  simul  anguH  majores  erunt  quatuor 
rectis,  propter  duos  additos  rectos  angulos  ad  punctum 
D.  Hoc  stante:  neutrius  modo  dictorum  triangulorum 
tres  simul  anguH  aequales  praecise  erunt,  aut  minores  duo- 
bus  rectis;  nam  sic  viceversa  alterius  trianguH  tres  simul 
anguH  majores  forent  duobus  rectis.  Quare  (ex  nona  hu- 
jus)  ab  uno  quidem  triangulo  stabiHretur  hypothesis  aut 
anguH  recti,  aut  anguH  acuti,  et  ab  altero  hypothesis  an- 
guH  obtusi,  quod  repugnat  quintae,  sextae,  et  septimae 
hujus.  Igitur  tres  simul  anguli  utriusque  praedictorum 
triangulorum  majores  erunt  duobus  rectis;  ac  propterea 
(ex  nona  hujus)  stabilietur  hypothesis  anguli  obtusi. 
Quod  erat  secundo  loco  demonstrandum. 


all  the  angles  of  the  triangles  ADB,  CDB  will  be  together 
equal  to  four  right  angles,  because  of  the  two  additional 
right  angles  at  the  point  D.  This  holding  good,  now  of 
neither  of  the  said  triangles,  as  suppose  ADB,  will  the 
three  angles  together  be  less,  or  greater  than  two  right 
angles;  for  thus  vice  versa  the  three  angles  together  of 
the  other  triangle  would  be  greater,  or  less  than  two  right 
angles.  Wherefore  (P.  IX.)  from  one  triangle  would 
indeed  be  established  the  hypothesis  of  acute  angle,  and 
from  the  other  the  hypothesis  of  obtuse  angle ;  [21]  which 
is  contrary  to  P.  VI.  and  P.  VII. 

Therefore  the  three  angles  together  of  either  of  the 
af oresaid  triangles  will  be  equal  to  two  right  angles ;  and 
thereby  (P.  IX.)  is  established  the  hypothesis  of  right 
angle. 

Quod  erat  primo  loco  demonstrandum. 

But  if  however  the  three  angles  of  the  proposed  tri- 
angle  ABC  are  taken  greater  than  two  right  angles :  now 
of  the  two  triangles  ADB,  CDB  all  the  angles  together 
will  be  greater  than  four  right  angles,  because  of  the  two 
additional  right  angles  at  the  point  D. 

This  holding  good:  now  of  neither  of  the  said  tri- 
angles  will  the  three  angles  together  be  precisely  equal 
to,  or  less  than  two  right  angles:  for  thus  vice  versa 
the  three  angles  of  the  other  triangle  would  be  together 
greater  than  two  right  angles.  Wherefore  (P.  IX.) 
f  rom  one  triangle  indeed  would  be  established  the  hypoth- 
esis  either  of  right  angle  or  of  acute  angle,  and  from  the 
other  the  hypothesis  of  obtuse  angle,  which  is  contrary 
to  Propp.  V.,  VL,  and  VIL 

Therefore  the  three  angles  together  of  either  of  the 
af oresaid  triangles  will  be  greater  than  two  right  angles ; 
and  therefore  is  established  the  hypothesis  of  obtuse 
angle. 

Quod  erat  secundo  loco  demonstrandum. 

63 


Tandem  vero.  Si  tres  anguli  propositi  trianguli  ABC 
ponantur  minores  duobus  rectis,  jam  duorum  triangulo- 
rum  ADB,  CDB,  omnes  simul  anguli  minores  erunt  qua- 
tuor  rectis,  propter  duos  additos  rectos  angulos  ad  punc- 
tum  D.  Hoc  stante:  neutrius  modo  dictorum  triangu- 
lorum  tres  simul  anguli  aequales  erunt,  aut  majores  duo- 
bus  rectis ;  nam  sic  viceversa  alterius  trianguli  tres  simul 
anguli  minores  forent  duobus  rectis.  Quare  (ex  nona 
hujus)  ab  uno  quidem  triangulo  stabiliretur  hypothesis 
aut  anguli  recti,  aut  anguli  obtusi,  et  ab  altero  hypothesis 
anguli  acuti;  quod  repugnat  quintae,  sextae,  et  septimae 
hujus.  Igitur  tres  simul  anguli  utriusque  praedictorum 
triangulorum  minores  erunt  duobus  rectis;  ac  propterea 
(ex  [22]  nona  hujus)  stabilietur  hypothesis  anguli  acuti. 
Quod  erat  tertio  loco  demonstrandum. 

Itaque  ex  quolibet  triangulo  ABC,  cujus  tres  simul 
anguli  aequales  sint,  aut  majores,  aut  minores  duobus  rec- 
tis,  stabilitur  respective  hypothesis  aut  anguli  recti,  aut 
anguli  obtusi,  aut  anguli  acuti.    Quod  erat  propositum. 

COROLLARIUM. 

Hinc;  protracto  uno  quolibet  cujusvis  propositi  trian- 
guli  latere,  ut  puta  AB  in  H;  erit  (ex  13.  primi)  externus 
angulus  HBC  aut  aequalis,  aut  minor,  aut  major  reliquis 
simul  internis,  et  oppositis  angulis  ad  puncta  A,  et  C, 
prout  vera  fuerit  hypothesis  aut  anguli  recti,  aut  anguli 
obtusi,  aut  anguli  acuti.    Et  vicissim. 


But  iinally.  If  the  three  angles  of  the  proposed 
triangle  ABC  are  taken  less  than  two  right  angles,  now 
of  the  two  triangles  ADB,  CDB,  all  the  angles  together 
will  be  less  than  four  right  angles,  because  of  the  two 
additional  right  angles  at  the  point  D. 

This  holding  good:  now  of  neither  of  the  said  tri- 
angles  will  the  three  angles  together  be  equal  to,  or 
greater  than  two  right  angles ;  for  thus  vice  versa  of  the 
other  triangle  the  three  angles  together  would  be  less 
than  two  right  angles. 

Wherefore  (P.  IX.)  from  one  triangle  indeed  would 
be  established  the  hypothesis  either  of  right  angle  or  of 
obtuse  angle,  and  from  the  other  the  hypothesis  of  acute 
angle;  which  is  contrary  to  Propp.  V.,  VI.,  and  VII. 

Therefore  the  three  angles  together  of  either  of  the 
aforesaid  triangles  will  be  less  than  two  right  angles ;  and 
therefore  (P.  IX.)  [22]  is  established  the  hypothesis  of 
acute  angle. 

Quod  erat  tertio  loco  demonstrandum. 

Accordingly  by  any  triangle  ABC,  of  which  the  three 
angles  are  together  equal  to,  or  greater,  or  less  than  two 
right  angles,  is  established  respectively  the  hypothesis  of 
right  angle,  or  obtuse  angle,  or  acute  angle. 

Quod  erat  propositum. 

COROLLARY. 

Hence,  any  one  side  of  any  proposed  triangle  being 
produced,  as  suppose  AB  to  H ;  the  external  angle  HBC 
will  be  (Eu.  I.  13)  equal  to,  or  less,  or  greater  than  the 
remaining  internal  and  opposite  angles  together  at  the 
points  A,  and  C,  according  as  is  true  the  hypothesis  of 
right  angle,  or  obtuse  angle,  or  acute  angle.  And  in- 
versely. 


65 


PROPOSITIO  XVI. 

Ex  quolibet  quadrilatero  ABCD,  cujus  quatuor  simul  an- 
guli  aequales  sint,  aut  majoreSj  aut  minores  quatuor 
rectiSj  stahilitur  respective  hypothesis  aut  anguli  recti, 
aut  anguli  obtusi,  aut  anguli  acuti. 

Demonstratur.  Jungatur  AC.  Non  erunt  (fig.  14.) 
tres  simul  anguli  trianguli  ABC  aequales,  aut  majores, 
aut  minores  duobus  rectis,  quin  tres  simul  anguli  trianguli 
ADC  sint  ipsi  etiam  respective  aequales,  aut  majores,  aut 
minores  duobus  rectis;  ne  scilicet  (ex  praecedente)  ab 
uno  illorum  triangulorum  stabiliatur  una  hypothesis,  et 
ab  altero  altera,  contra  quintam,  sextam,  et  septimam 
hujus.  Hoc  stante:  Si  quatuor  simul  anguli  propositi 
quadrilateri  aequales  sint  quatuor  rectis,  constat  utrius- 
que  modo  dictorum  triangulorum  tres  simul  angulos 
aequales  fore  duobus  rectis,  atque  ideo  (ex  praecedente) 
stabiH-[23]tum  iri  hypothesim  anguH  recti. 

Sin  vero  ejusdem  quadrilateri  quatuor  simul  anguli 
majores  sint,  aut  minores  quatuor  rectis,  debebunt  simili- 
ter  illorum  triangulorum  tres  simul  anguH  respective  esse 
aut  una  majores,  aut  una  minores  duobus  rectis.  Quare 
ab  ihis  trianguHs  stabiHetur  respective  (ex  praecedente) 
aut  hypothesis  anguH  obtusi,  aut  hypothesis  anguH  acuti. 

Itaque  ex  quoHbet  quadrilatero,  cujus  quatuor  simul 
anguH  aequales  sint,  aut  majores,  aut  minores  quatuor 


66 


PROPOSITION  XVI. 

By  any  quadrilateral  ABCD,  of  which  the  four  angles 
together  are  equal  to,  or  greater,  or  less  than  four 
right  angles,  is  established  respectively  the  hypothesis 
of  right  angle,  or  obtuse  angle,  or  acute  angle. 

Proof.  Join  AC.  The  three  angles  of  the  triangle 
ABC  (fig.  14)  will  not  be  together  equal  to,  or  greater, 
or  less  than  two  right  angles, 
without  the  three  angles  of  the 
triangle  ADC  being  themselves 
also  together  respectively  equal 
to,  or  greater,  or  less  than  two 
right  angles,  lest  obviously  (by 
the  preceding)  from  one  of 
those  triangles  be  established  one  hypothesis,  and  another 
f rom  the  other,  against  the  fifth,  sixth,  and  seventh  prop- 
ositions  of  this  work. 

This  holding  good :  If  the  four  angles  together  of  the 
premised  quadrilateral  are  equal  to  four  right  angles,  it 
follows  that  the  three  angles  together  of  either  of  the 
just  mentioned  triangles  will  be  equal  to  two  right  angles, 
and  therefore  (from  the  preceding)  ['^^]  the  hypothesis 
of  right  angle  will  be  established. 

But  if  indeed  the  four  angles  of  this  quadrilateral  be 
together  greater,  or  less  than  four  right  angles,  similarly 
the  three  angles  together  of  those  triangles  should  be 
respectively  either  at  the  same  time  greater,  or  at  the  same 
time  less  than  two  right  angles.  Wherefore  from  these 
triangles  would  be  established  respecti vely  ( f  rom  the  pre- 
ceding)  either  the  hypothesis  of  obtuse  angle,  or  the  hy- 
pothesis  of  acute  angle. 

Therefore  by  any  quadrilateral,  of  which  the  four 
angles  together  are  equal  to,  or  greater,  or  less  than  four 

67 


rectis,  stabilitur  respective  hypothesis  aut  anguli  recti, 
aut  anguli  obtusi,  aut  anguli  acuti.  Quod  erat  demon- 
strandum. 

COROLLARIUM. 

Hinc :  protractis  versus  easdem  partes  duobus  quibus- 
vis  propositi  quadrilateri  contrapositis  lateribus,  ut  puta 
AD  in  H,  et  BC  in  M;  erunt  (ex  13.  primi)  duo  simul 
externi  anguli  HDC,  MCD  aut  aequales,  aut  minores, 
aut  majores  duobus  simul  internis,  et  oppositis  anguHs 
ad  puncta  A,  et  B,  prout  vera  fuerit  hypothesis  aut  an- 
guH  recti,  aut  anguli  obtusi,  aut  anguH  acuti. 

PROPOSITIO  XVIL 

Si  uni,  ut  lihet,  cuidam  parvae  rectae  AB  insistat  (fig.  15.) 
ad  rectos  angulos  recta  AH:  Dico  subsistere  non 
posse  in  hypothesi  anguli  acuti,  ut  quaevis  BD,  effi- 
ciens  cum  AB  quemlihet  angulum  acutum  versus 
partes  ipsiiis  AH,  occursura  tandeni  sit  ad  finitam, 
seu  terminatam  distantiam  ipsi  AH  productae. 

Demonstratur.  Jungatur  HB.  Erit  (ex  17.  primi) 
acutus  angulus  ABH,  propter  angulum  rectum  ad  punc- 
tum  A.  Jam  (ex  23.  primi)  ducatur  quaedam  HD  ver- 
[24]sus  partes  puncti  B,  quae  non  secans  angulum  AHB 
efficiat  cum  ipsa  HB  angulum  acutum  aequalem  ipsi  acuto 
ABH.  Deinde  ex  puncto  B  demittatur  ad  HD  perpen- 
dicularis  BD,  quae  cadet  ad  partes  praedicti  anguli  acuti 
ad  punctum  H.  Ouoniam  igitur  latus  HB  opponitur  in 
triangulo  HDB  angulo  recto  in  D,  atque  item  in  triangulo 


68 


right  angles,  is  established  respectively  the  hypothesis  of 
right  angle,  or  obtuse  angle,  or  acute  angle. 
Quod  erat  demonstrandum. 

COROLLARY. 

Hence,  any  two  opposite  sides  of  the  premised  quadri- 
lateral  being  produced  toward  the  same  parts,  as  suppose 
AD  to  H,  and  BC  to  M;  the  two  external  angles  HDC, 
MCD  will  be  (Eu.  L  13)  either  equal  to,  or  less,  or 
greater  than  the  two  internal  and  opposite  angles  together 
at  the  points  A,  and  B,  according  as  is  true  the  hypothesis 
of  right  angle,  or  obtuse  angle,  or  acute  angle. 

PROPOSITION  XVIL 

//  the  straight  AH  stands  (fig.  15)  at  right  angles  to 
any  certain  arbitrarily  small  straight  AB:  I  say  that 
in  the  hypothesis  of  acute  angle  it  cannot  hold  good, 
that  every  straight  BD,  making  with  AB  tozvard  the 
parts  of  this  AH  any  acufe  angle 
you  choose,  will  at  length  meet 
this  AH  produced  at  a  finite,  or 
terminated  distance. 


Proof.  JoinHB.  The  angle  ABH 
will  be  acute  (Eu.  I.  17)  because  of 
the  right  angle  at  the  point  A.  Now 
draw  (Eu.  L  23)  HD  toward  [24]  the  ^-^^^ 

parts  of  the  point  B,  which  not  cut- 
ting  the  angle  AHB  makes  with  this  HB  an  acute  angle 
equal  to  this  acute  angle  ABH.  Then  from  the  point 
B  is  let  fall  to  HD  the  perpendicular  BD,  which  will 
fall  toward  the  parts  of  the  aforesaid  acute  angle  at  the 
point  H. 

Since  therefore  the  side  HB  is  opposite  in  the  triangle 
HDB  to  the  right  angle  at  D,  and  likewise  in  the  triangle 


69 


BAH  angulo  recto  in  A;  ac  rursum  in  duobus  illis  trian- 
gulis  adjacent  eidem  lateri  HB  aequales  anguli,  qui  sunt 
in  priore  quidem  triangulo  angulus  BHD,  et  in  posteriore 
angulus  HBA;  erit  etiam  (ex  26.  primi)  reliquus  angulus 
HBD  in  priore  triangulo  aequalis  reliquo  angulo  BHA  in 
posteriore  triangulo.  Quare  integer  angulus  DBA  aequa- 
lis  erit  integro  angulo  AHD. 

Jam  vero:  non  erit  uterque  praedictorum  aequalium 
angulorum  obtusus,  ne  incidamus  (ex  praecedente)  in 
unum  casum  jam  reprobatae  hypothesis  anguH  obtusi.  Sed 
neque  erit  rectus,  ne  incidamus  (ex  eadem  praecedente) 
in  unum  casum  pro  hypothesi  anguli  recti,  qui  nullum 
(ex  5.  hujus)  rehnqueret  locum  hypothesi  anguli  acuti. 
Uterque  igitur  illorum  angulorum  erit  acutus.  Hoc  stante : 
Quod  recta  BD  protracta  occurrere  nequeat  in  quodam 
puncto  K  ipsi  AH  ad  easdem  partes  productae,  ex  eo 
demonstratur ;  quia  in  triangulo  KDH,  praeter  angulum 
rectum  in  D,  adesset  angulus  obtusus  in  H,  cum  angulus 
AHD,  in  praedicta  hypothesi  anguli  acuti,  demonstratus 
sit  acutus.  Hoc  autem  absurdum  est,  contra  17.  primi. 
Non  ergo  subsistere  potest  in  ea  hypothesi,  ut  quaevis  BD, 
efficiens  cum  una,  ut  libet  parva  recta  AB,  quemlibet  an- 
gulum  acutum  versus  partes  ipsius  AH,  occursura  tan- 
dem  sit  ad  finitam,  seu  terminatam  distantiam,  ipsi  AH 
productae.     Quod  erat  demonstrandum. 

Aliter  idem,  ac  facilius.  Insistant  uni  cuidam  quan- 
tumlibet  parvae  rectae  AB  (fig.  16.)  duae  perpendiculares 
[25]  AK,  BM.  Demittatur  ad  AK  ex  aliquo  puncto  M 
ipsius  BM  perpendicularis  MH,  jungaturque  BH.  Con- 
stat  acutum  fore  angulum  BHM.    Est  etiam  (ex  praece- 


70 


BAH  to  the  right  angle  at  A;  and  again  in  those  two 
triangles  equal  angles  are  adjacent  to  this  side  HB,  which 
are  in  the  first  triangle  indeed  the  angle  BHD,  and  in  thc 
latter  the  angle  HBA;  also  (Eu.  I.  26)  the  remaining 
angle  HBD  in  the  former  triangle  will  be  equal  to  the 
remaining  angle  BHA  in  the  latter  triangle.  Wherefore 
the  entire  angle  DBA  will  be  equal  to  the  entire  angle 
AHD. 

Now  however,  neither  of  the  aforesaid  equal  angles 
will  be  obtuse,  lest  we  meet  (from  the  preceding  propo- 
sition)  a  case  of  the  now  rejected  hypothesis  of  obtuse 
angle. 

Nor  will  either  be  right,  lest  we  meet  (from  the 
same  preceding)  a  case  of  the  hypothesis  of  right  angle, 
which  (P.  V.)  will  leave  no  place  for  the  hypothesis  of 
acute  angle.  Therefore  each  one  of  those  angles  will  be 
acute.  This  being  the  case:  that  the  straight  BD  pro- 
duced  cannot  meet  in  a  certain  point  K  this  AH  produced 
toward  the  same  parts,  is  demonstrated  thus;  because  in 
the  triangle  KDH,  besides  the  right  angle  at  D,  is  present 
the  obtuse  angle  at  H,  since  the  angle  AHD  in  the  afore- 
said  hypothesis  of  acute  angle  is  proved  acute.  But  this 
is  absurd,  against  Eu.  I.  17. 

Therefore  it  cannot  hold  good  in  this  hypothesis,  that 
any  BD,  making  with  an  arbitrarily  small  straight  AB 
any  acute  angle  toward  the  parts  of  this  AH,  will  at 
length  at  a  finite,  or  terminated  distance,  meet  this  AH 
produced. 

Quod  erat  demonstrandum. 

The  same  otherwise  and  more  easily. 

Two  perpendiculars  AK,  BM  stand  on  a  certain 
straight  AB,  as  small  as  you  choose  (fig.  16).  [25]  From 
any  point  M  of  this  BM  let  fall  to  AK  the  perpendicular 
MH,  and  join  BH.    It  follows  that  the  angle  BHM  will 


71 


dente)  acutus  angulus  BMH,  in  hypothesi  anguli  acuti. 
Ergo  perpendicularis  BDX,  ex  puncto  B  ad  ipsam  HM 
demissa,  secabit  (ex  17.  prinii)  eam  HM  in  quodam 
puncto  intermedio  D.  Ergo  angulus  XBA  erit  acutus. 
Constat  autem  (ex  eadem  17.  primi)  non  posse  invicem 
concurrere  (saltem  ad  finitam,  seu  terminatam  distan- 
tiam)  duas  illas  utcunque  productas  AHK,  BDX,  propter 
angulos  rectos  in  punctis  H,  et  D.  Itaque  nequit  subsis- 
tere  in  hypothesi  anguli  acuti,  ut  quaevis  BD,  efficiens 
cum  una,  ut  Hbet,  parva  recta  AB,  quemlibet  angulum 
acutum  versus  partes  ipsius  AH,  eidem  AB  perpendicu- 
laris,  occursura  tandem  sit  (ad  finitam,  seu  terminatam 
distantiam)  ipsi  AH  productae.     Quod  erat  propositum. 

SCHOLION  I. 

Atque  id  est,  quod  spopondi  in  Scholiis  post  XHI.  hu- 
jus,  nimirum  destructum  iri  hypothesim  anguH  acuti  (quae 
sola  obesse  jam  potest  generaH  ihi  Pronunciato  EucH- 
daeo)  ex  solo  admisso  generaH  duarum  rectarum  con- 
cursu  ad  eas  partes,  versus  quas  recta  quaepiam,  quan- 
tumHbet  parva,  in  easdem  incidens,  duos  efficiat  internos 
angulos  minores  duobus  rectis ;  atque  ita  quidem,  etiamsi 
alteruter  illorum  angulorum  supponi  debeat  rectus. 

SCHOLION   IL 

Sed  rursum  meHore  loco,  post  XXVII.  hujus,  ostendam 
destructum  pariter  iri  hypothesim  anguH  acuti,  dum  unus 
aHquis  tenuissimus,  ut  Hbet,  angulus  acutus  desi-[26]gnari 


12 


Fig.  16. 


be  acute.     In  the  hypothesis  of  acute  angle,  the  angle 

BMH   is  also    (from  the  preceding 

proposition)    acute.     The-refore  the 

perpendicular  BDX,  let  fall  from  the 

point  B   to  this   HM,   will  cut    (by 

Eu.  I.   17)   this  HM  in  some  inter- 

mediate  point  D.  Therefore  the  angle 

XBA  will  be  acute. 

But  it  follows  (Eu.  I.  17)  that 
those  two  straights  AHK,  BDX  how- 
ever  produced  cannot  meet  (anyhow  at  a  finite  or  ter- 
minated  distance)  on  account  of  the  right  angles  at  the 
points  H  and  D.  Therefore  in  the  hypothesis  of  acute 
angle  it  cannot  hold  good,  that  any  BD,  making  with 
a  straight  AB,  however  small,  any  acute  angle  toward 
the  parts  of  this  AH,  perpendicular  to  this  same  AB, 
will  at  length  meet  (at  a  finite  or  terminated  distance) 
this  AH  produced. 

Quod  erat  propositum. 

SCHOLION  I. 

And  this  is  what  I  promised  in  the  scholia  after 
P.  Xni.,  that  the  hypothesis  of  acute  angle  (which  alone 
is  able  now  to  stand  against  that  general  EucHdean  as- 
sumption)  will  certainly  be  destroyed  by  the  sole  ad- 
mission  of  a  universal  meeting  of  two  straights  toward 
those  parts  toward  which  any  straight,  as  small  as  you 
choose,  meeting  them,  makes  two  internal  angles  less  than 
two  right  angles;  and  just  so,  even  if  either  of  those 
angles  is  to  be  supposed  right. 

SCHOLION  IL 

But  again  in  a  better  place,  after  P.  XXVII.,  I  shall 
show  that  the  hypothesis  of  acute  angle  will  be  equally 
destroyed,  provided  that  any  one  acute  angle  as  small  as 

73 


possit;  sub  quo,  si  recta  quaepiam  in  alteram  incidat, 
debeat  haec  producta  (ad  finitam,  seu  terminatam  distan- 
tiam)  aliquando  occurrere  cuivis  ad  quantamlibet  finitam 
distantiam  excitatae  super  ea  incidente  perpendiculari. 

PROPOSITIO  XVIII. 

Ex  quolibet  triangulo  ABC,  cujus  angulus  (fig.  17.)  ad 
punctum  B  in  uno  quovis  semicirculo  existat,  cujus 
diameter  AC,  stahilitur  hypothesis  aut  anguli  recti, 
aut  anguli  ohtusi,  aut  anguli  acuti,  prout  nempe  an- 
gulus  ad  punctum  B  fuerit  aut  rectus,  aut  ohtusus, 
aut  acutus. 

Demonstratur.  Ex  centro  D  jungatur  DB.  Erunt 
(ex  quinta  primi)  aequales  anguli  ad  basim  AB,  atque 
item  ad  basim  BC,  in  triangulis  ADB,  CDB.  Quare,  in 
triangulo  ABC,  duo  simul  anguli  ad  basim  AC  aequales 
erunt  toti  angulo  ABC.  Igitur  tres  simul  anguli  trianguli 
ABC  aequales  erunt,  aut  majores,  aut  minores  duobus 
rectis,  prout  angulus  ad  punctum  B  fuerit  aut  rectus,  aut 
obtusus,  aut  acutus.  Itaque  ex  quolibet  triangulo  ABC, 
cujus  angulus  ad  punctum  B  in  uno  quovis  semicirculo 
existat,  cujus  diameter  AC,  stabilitur  (ex  15.  hujus)  hy- 
pothesis  aut  anguH  recti,  aut  anguH  obtusi,  aut  anguH 
acuti,  prout  nempe  angulus  ad  punctum  B  fuerit  aut  rec- 
tus,  aut  obtusus,  aut  acutus.    Quod  erat  etc. 

PROPOSITIO  XIX. 

Esto  quodvis  triangulum  AHD  (fig.  18.)  rectangulum  in 
H.  Tum  in  AD  continuata  sumatur  portio  DC 
aequalis  ipsi  AD;  demittaturque  ad  AH  produ^tam 


74 


you  choose  can  be  designated,[26]  under  which  if  any 
straight  Hne  meets  another,  this  produced  must  (at  a 
finite  or  terminated  distance)  finally  meet  any  perpen- 
dicular  erected  upon  this  incident  straight  at  whatever 
finite  distance. 

PROPOSITION  XVIII. 

From  any  triangle  ABC,  of  which  (fig.  17)  the  angle  at 
the  point  B  is  inscrihed  in  any  semicircle  of  diameter 
AC,  is  established  the  hypothesis  of  right  angle,  or 
obtuse  anglej  or  acute  angle,  according  as  indeed  the 
angle  at  the  point  B  is  right,  or  obtuse,  or  acute. 

Proof.  From  the  center  D  join  DB.  The  angles  at 
the  base  AB  will  be  (Eu.  I.  5)  equal,  and  Hkewise  at 
the  base  BC,  in  the  triangles  ADB, 
CDB.  Wherefore,  in  the  triangle 
ABC  the  two  angles  at  the  base 
AC  will  be  together  equal  to  the 
whole  angle  ABC.  Therefore  the 
three  angles  of  the  triangle  ABC 
will  be  together  equal  to,  or  greater,  or  less  than  two 
right  angles,  according  as  the  angle  at  the  point  B  is  right, 
or  obtuse,  or  acute. 

Therefore  from  any  triangle  ABC,  of  which  the  angle 
at  the  point  B  is  inscribed  in  any  semicircle  of  diameter 
AC,  is  estabHshed  (P.  XV.)  the  hypothesis  of  right  angle, 
or  obtuse  angle,  or  acute  angle,  according  as  indeed  the 
angle  at  the  point  B  is  right,  or  obtuse,  or  acute. 

Quod  erat  demonstrandum. 

PROPOSITION  XIX. 

Let  there  be  any  triangle  AHD  (fig.  18)  right-angled  at 
H.  Then  in  AD  produced  the  portion  DC  is  as- 
sumed  equal  to  this  AD;  and  the  perpendicular  CB 

75 


perpendictdaris  CB.  Dico  stabilitum  hinc  iri  hypo- 
thesim  aut  anguli  recti,  aut  anguli  obtusi,  aut  anguli 
acuti,  prout portio  HB  aequalis  fue-i^^^rit,  aut  major, 
aut  minor  ipsa  AH. 

Demonstratur.  Nam  juncta  DB  erit  (ex  4.  primi,  et 
ex  10.  hujus)  aut  aequalis,  aut  major,  aut  minor  ipsa 
AD,  sive  DC,  prout  illa  portio  HB  aequalis  fuerit,  aut 
major,  aut  minor  ipsa  AH. 

Et  primo  quidem  sit  HB  aequalis  ipsi  AH,  ita  ut 
propterea  juncta  DB  aequalis  sit  ipsi  AD,  sive  DC.  Con- 
stat  circumferentiam  circuli,  qui  centro  D,  et  intervallo 
DB  describatur,  transituram  per  puncta  A,  et  C.  Igitur 
angulus  ABC,  qui  ponitur  rectus,  existet  in  eo  semicir- 
culo,  cujus  diameter  AC.  Quare  (ex  praecedente)  sta- 
bilietur  hypothesis  anguli  recti.  Quod  erat  primo  loco 
demonstrandum. 

Sit  secundo  HB  major  ipsa  AH,  ita  ut  propterea 
juncta  DB  major  sit  ipsa  AD,  sive  DC.  Constat  circum- 
ferentiam  circuli,  qui  centro  D,  et  intervallo  DA,  sive  DC, 
describatur,  occursuram  ipsi  DB  in  ahquo  puncto  inter- 
medio  K.  Igitur,  junctis  AK,  et  CK,  erit  angulus  AKC 
obtusus,  quia  major  (ex  21.  primi)  angulo  ABC,  qui 
ponitur  rectus.  Quare  (ex  praecedente)  stabilietur  hypo- 
thesis  anguli  obtusi.  Quod  erat  secundo  loco  demon- 
strandum. 

Sit  tertio  HB  minor  ipsa  AH,  ita  ut  propterea  juncta 
DB  minor  sit  ipsa  AD,  sive  DC.  Constat  circumferen- 
tiam  circuH,  qui  centro  D,  et  intervallo  DA,  sive  DC 
describatur,  occursuram  in  aHquo  puncto  M  ipsius  DB 


76 


is  let  fall  to  AH  prodnced.  I  say  hence  will  be 
established  the  hypothesis  of  right  angle,  or  obtuse 
angle,  or  acute  angle,  according  as  the  portion  HB 
is  eqtial  to,  [27]  or  greater,  or  less  than  AH. 

Proof.  For  the  join  DB  will  be  (Eu.  I.  4,  and  P.  X. 
of  this)  either  equal  to,  or  greater,  or  less  than  AD,  or 
DC,  according  as  the  portion  HB 
is  equal  to,  or  greater,  or  less  than 
AH. 

And  first  indeed  let  HB  be  equal 
to  AH,  so  that  therefore  the  join 
DB  may  be  equal  to  AD,  or  DC.  It 
follows  that  the  circumference  of 
the  circle,  which  is  described  with 
the  center  D  and  radius  DB,  will  '^' 

go  through  the  points  A  and  C.  Therefore  the  angle 
ABC,  which  is  assumed  right,  is  in  this  semicircle,  whose 
diameter  is  AC.  Wherefore  (from  the  preceding  propo- 
sition)  is  estabHshed  the  hypothesis  of  right  angle. 

Quod  erat  primo  loco  demonstrandum. 

Secondly  let  BH  be  greater  than  AH,  so  that  there- 
fore  the  join  DB  is  greater  than  AD,  or  DC.  It  follows 
that  the  circumference  of  the  circle,  which  is  described 
with  center  D,  and  radius  DA,  or  DC,  will  meet  DB  in 
some  intermediate  point  K.  Therefore,  AK,  and  CK 
being  joined,  the  angle  AKC  will  be  obtuse,  because 
greater  (Eu.  I.  21)  than  the  angle  ABC,  which  is  as- 
sumed  right.  Wherefore  (from  the  preceding  proposi- 
tion)  is  established  the  hypothesis  of  obtuse  angle. 

Quod  erat  secundo  loco  demonstrandum. 

Thirdly  let  BH  be  less  than  AH,  so  that  therefore 
the  join  DB  is  less  than  AD,  or  DC.  It  follows  that  the 
circumference  of  the  circle,  which  is  described  with  cen- 
ter  D,  and  radius  DA,  or  DC,  will  meet  in  some  point  M 


77 


ulterius  protractae.  Igitur  junctis  AM,  et  CM,  erit  an- 
gulus  AMC  acutus,  quia  minor  (ex  eadem  21.  primi) 
illo  angulo  ABC,  qui  ponitur  rectus.  Quare  (ex  praece- 
dente)  stabilietur  hypothesis  anguH  acuti.  Quod  erat 
tertio  loco  demonstrandum.  Itaque  constant  omnia  pro- 
posita.  [28] 

PROPOSITIO  XX. 

Esto  triangulum  ACM  (fig.  19.)  rectangulum  in  C.  Tum 
ex  puncto  B  dividente  hifariam  ipsam  AM  demit- 
tatur  ad  AC  perpendicularis  BD.  Dico  hanc  per- 
pendicularem  majorem  non  fore  (in  hypothesi  anguli 
acuti)  medietate  perpendicularis  MC. 

Demonstratur.  Continuetur  enim  DB  usque  ad  DH 
duplam  ipsius  DB.  Foret  igitur  DH  (si  DB  major  sit 
praedicta  medietate)  major  ipsa  CM,  ac  propterea  aequaHs 
cuidam  continuatae  CMK.  Jungantur  AH,  HK,  HM, 
MD.  Jam  sic  progredimur.  Quoniam  in  trianguHs  HBA, 
DBM,  aequaHa  ponuntur  latera  HB,  BA,  lateribus  DB, 
BM;  suntque  (ex  15.  primi)  aequales  anguH  ad  punctum 
B;  erit  etiam  (ex  quarta  ejusdem  primi)  basis  HA  aequa- 
Hs  basi  MD.  Deinde,  propter  eandem  rationem,  aequales 
erunt  in  trianguHs  HBM,  DBA,  bases  HM,  DA.  Quare 
in  trianguHs  MHA,  ADM,  aequales  erunt  (ex  8.  primi) 
anguH  MHA,  ADM.  Rursum  in  trianguHs  AHB,  MDB, 
aequaHs  manebit  angulus  residuus  MHB  residuo  recto 
angulo  ADB.     Igitur  rectus  erit  angulus  MHB.     At  hoc 


this  DB  produced  outwardly.  Therefore  AM  and  CM 
being  joined,  the  angle  AMC  will  be  acute,  because  less 
(Eu.  I.  21)  than  the  angle  ABC,  which  is  assumed  right. 

Therefore  (from  the  preceding  proposition)  is  estab- 
lished  the  hypothesis  of  acute  angle. 

Quod  erat  tertio  loco  demonstrandum. 

Itaque  constant  omnia  proposita.  [28] 


PROPOSITION  XX. 

Let  there  be  a  triangle  ACM  (fig.  19)  right-angled  at  C. 
Then  from  the  point  B  bisecting  this  AM  let  fall  the 
perpendicular  BD  to  AC.  I  say  this  perpendicular 
will  not  be  (in  the  hypothesis  of  acute  angle)  greater 
than  half  the  perpendicular  MC. 

Proof.  For  let  DB  be  produced  to  DH  double  DB, 
Therefore  DH  would  be  (if  DB  be 
greater  than  the  aforesaid  half) 
greater  than  CM,  and  therefore 
equal  to  a  certain  continuation 
CMK. 

Join  AH,  HK,  HM,  MD.  Now 
we  proceed  thus.  Since  in  the  tri- 
angles  HBA,  DBM,  the  sides  HB, 
BA  are  assumed  equal  to  the  sides 
DB,  BM;  and  (Eu.  I.  15)  the 
angles  at  the  point  B  are  equal; 
the  base  HA  also  (Eu.  I.  4)  will  be  equal  to  the  base 
MD.  Then,  by  the  same  reasoning,  in  the  triangles 
HBM,  DBA,  the  bases  HM,  DA  will  be  equal.  Where- 
fore  in  the  triangles  MHA,  ADM,  the  angles  MHA, 
ADM  (Eu.  I.  8)  will  be  equal. 

Again  in  the  triangles  AHB,  MDB,  the  residual  angle 
MHB  will  remain  equal  to  the  residual  right  angle  ADB. 
Therefore  the  angle  MHB  will  be  right.    But  this  is  ab- 


Fig.  19. 


absurdum  est,  in  hypothesi  anguli  acuti;  cum  recta  KH 
jungens  aequaHa  perpendicula  KC,  HD,  acutos  angulos 
efficiat  cum  eisdem  perpendicuHs.  Non  ergo  perpendicu- 
laris  BD  major  est  (in  hypothesi  anguH  acuti)  medietate 
perpendicularis  MC.     Quod  erat  demonstrandum. 

PROPOSITIO  XXI. 
lisdem  manentibtis:  Intelligantur  in  infinitum  produci 
ipsae  AM,  et  AC.  Dico  earundem  distantiam  majo- 
rem  fore  (in  utraque  hypothesi  aut  angidi  recti,  aiit 
anguli  acuti)  qualibet  assignabili  finita  longitudine. 
[29] 

Demonstratur.  In  AM  continuata  sumatur  AP  dupla 
ipsius  AM,  demittaturque  ad  AC  continuatam  perpendi- 
cularis  PN.  Non  erit  (ex  praecedente)  in  utravis  prae- 
dicta  hypothesi  perpendicularis  MC  major  medietate  per- 
pendicularis  PN.  Igitur  PN  saltem  erit  dupla  ipsius  MC, 
prout  MC  saltem  est  dupla  alterius  BD.  Atque  ita  sem- 
per,  si  in  continuata  AM  sumatur  dupla  ipsius  AP,  ex 
ejusque  termino  demittatur  perpendicularis  ad  continua- 
tam  AC.  Scilicet  perpendicularis,  quae  ex  AM  semper 
magis  continuata  demittetur  ad  continuatam  AC,  multi- 
plex  erit  determinatae  BD  supra  quemlibet  finitum  assig- 
nabilem  numerum.  Igitur  praedictarum  rectarum  dis- 
tantia  major  erit  (in  utraque  praedicta  hypothesi)  quali- 
bet  assignabili  iinita  longitudine.  Quod  erat  demon- 
strandum. 

COROLLARIUM. 

Quoniam  vero  hypothesis  anguli  obtusi,  quae  unice 
obesse  hic  posset,  demonstrata  jam  est  absolute  falsa; 


80 


surd  in  the  hypothesis  of  acute  angle;  since  the  straight 
KH  joining  equal  perpendiculars  KC,  HD,  makes^  acute 
angles  with  these  perpendiculars. 

Therefore  the  perpendicular  BD  is  not  (in  the  hypoth- 
esis  of  acute  angle)  greater  than  the  half  of  the  perpen- 
dicular  MC. 

Quod  erat  demonstrandum. 

PROPOSITION  XXI. 

The  same  remaining :  If  AM  and  AC  are  understood  as 
produced  in  infinitum  I  say  their  distance  (in  either 
the  hypothesis  of  right  angle,  or  of  acute  angle)  will 
he  greater  than  any  assignahle  finite  length.  [29] 

Proof.  In  AM  produced  assume  AP  double  of  AM, 
and  let  fall  to  AC  produced  the  perpendicular  PN. 

The  perpendicular  MC  will  not  be  (from  the  pre- 
ceding)  in  either  hypothesis  aforesaid  greater  than  half 
the  perpendicular  PN.  Therefore  PN  will  be  at  least 
double  MC,  just  as  MC  is  at  least  double  BD. 

And  so  always,  if  in  AM  produced  is  assumed  double 
AP,  and  from  the  terminus  of  this  a  perpendicular  is  let 
fall  to  AC  produced. 

It  is  obvious  that  the  perpendicular,  which  from  AM 
ever  more  produced  is  let  fall  to  AC  produced,  will  be 
a  multiple  of  the  determinate  BD  beyond  any  finite  as- 
signable  number. 

Therefore  the  distance  of  the  aforesaid  straights  will 
be  (in  either  aforesaid  hypothesis)  greater  than  any 
assignable  finite  length. 

Quod  erat  demonstrandum. 

COROLLARY. 

But  since  the  hypothesis  of  obtuse  angle,  which  alone 
could  hinder  here,  is  already  proved  absolutely  false; 
1  Propp.  L,  VIL.  and  XVI. 


consequitur  sane  absolute  verum  esse,  quod  distantia 
unius  ab  altera  praedictarum  rectarum,  si  in  infinitum 
producantur,  major  sit  qualibet  finita  assignabili  longi- 
tudine. 

SCHOLION  I. 

In  quo  expenditur  conatus  Procli. 

Post  Theoremata  a  me  huc  usque  demonstrata  sinc 
ulla  dependentia  ab  illo  Pronunciato  EucHdaeo,  ad  cujus 
nempe  exactissimam  demonstrationem  omnia  conspirant; 
operae  pretium  facturum  me  judico,  si  quorundam  etiam 
celebriorum  Geometrarum  labores  in  eandem  me-[30]tam 
contendentium  diligenter  expendam.  Incipio  a  Proclo, 
cujus  est  apud  Clavium  in  Elementis  post  XXVIII. 
Libri  primi  sequens  assumptum:  Si  ab  uno  puncto  duae 
rectae  lineae  angulum  facientes  infinite  producantur,  ipsa^ 
rum  distantia  omnem  finitam  magnitudinem  excedet.  At 
Proclus  demonstrat  quidem  (ut  ibi  optime  advertit  Cla- 
vius)  duas  rectas  (fig.  20.)  ut  puta  AH,  AD  ab  eodem 
puncto  A  exeuntes  versus  easdem  partes,  semper  magis, 
in  majore  distantia  ab  eo  puncto  A,  inter  se  distare,  sed 
non  etiam  ita  ut  ea  distantia  crescat  ultra  omnem  finitum 
designabilem  limitem,  prout  opus  foret  ad  ipsius  inten- 
tum.  Quo  loco  praefatus  Clavius  affert  exemplum  Con- 
choidis  Nicomedeae,  quae  cum  recta  AH  ex  eodem  puncto 
A  versus  easdem  partes  exiens,  ita  semper  magis  ab 
eadem  recedit,  ut  tamen  ipsarum  distantia  non  nisi  ad 
infinitam  earundem  productionem,  aequalis  sit  cuidam 
finitae  rectae  AB  perpendiculariter  insistenti  ipsis  AH, 
BC,  versus  easdem  partes  in  infinitum  protractis.     Quid 


so  of  course  follows  as  absolutely  true,  that  the  distance 
of  one  from  the  other  of  the  aforesaid  straights,  if  they 
be  produced  in  infinitum,  is  greater  than  any  finite  assign- 
able  length. 

SCHOLION  I. 

In  which  is  weighed  the  endeavor  of  Proclus. 

After  the  theorems  so  far  demonstrated  by  me,  inde- 
pendently  of  the  Euclidean  postulate,  toward  an  exact 
proof  of  which  they  all  conspire;  in  my  judgment  it  is 
well  if  I  diligently  weigh  the  labors  of  certain  well- 
known  geometers  in  the  same  endeavor.  1^0] 

I  begin  from  Proclus,  of  whom  Clavius  in  the  Ele- 
ments,  after  I.  28,  gives  the  following  assumption: 

//  frofn  a  point  two  straight  lines  making  an  angle 
are  produced  infinitely,  their  distance  will  exceed  every 
finite  magnitude. 

But  Proclus  demonstrates  indeed  (as  Clavius  there 
well  remarks)  that  two  straights  (fig.  20)  as  suppose 
AH,  AD  going  out  from  the 
same  point  A  toward  the  same 
parts,  always  diverge  the  more 
from  each  other,  the  greater  the 
distance   from  the  point  A,  but  Fig.  20. 

not    also    that   this    distance    in- 

creases  beyond  every  finite  limit  that  may  be  designated, 
as  was  requisite  for  his  purpose. 

In  which  place  the  aforesaid  Clavius  cites  the  example 
of  the  Conchoid  of  Nicomedes,  which  going  out  from 
the  same  point  A  as  the  straight  AH  toward  the  same 
parts,  so  recedes  always  more  from  it,  that  nevertheless 
only  at  an  infinite  production  is  their  distance  equal  to  a 
certain  finite  sect  AB  standing  perpendicular  to  AH  and 
BC  produced  in  infinitum  toward  the  same  parts.    Why 

«3 


ni  ergo,  nisi  specialis  ratio  in  contrarium  cogat,  dici  idem 
possit  de  duabus  suppositis  rectis  lineis  AH,  AD? 

Neque  hic  accusari  potest  Clavius,  quod  Proclo  op- 
ponat  eam  Conchoidis  proprietatem,  quae  nempe  demon- 
strari  non  potest  sine  adjumento  plurium  Theorematum, 
Pronunciato  hic  controverso  innixorum.  Nam  dico  ex 
hoc  ipso  confirmari  vim  redargutionis  Clavianae;  quia 
scilicet  ex  illo  Pronunciato  assumpto  ut  vero  manifeste 
consequitur,  duas  lineas  in  infinitum  protractas,  unam 
rectam,  et  alteram  inflexam,  posse  unam  ab  altera  semper 
magis  recedere  intra  quendam  finitum  determinatum  limi- 
tem ;  unde  utique  oriri  potest  suspicio,  ne  simile  quidpiam 
contingere  possit  in  duabus  lineis  rectis,  nisi  aliter  demon- 
stretur. 

Sed  non  idcirco;  postquam  ego  in  Cor.  praecedentis 
[31]  Propositionis  manifestam  jam  feci  absolutam  veri- 
tatem  praecitati  assumpti;  transiri  statim  potest  ad  asse- 
rendum  Pronunciatum  illud  EucHdaeum.  Nam  antea 
demonstrari  etiam  oporteret,  quod  duae  illae  rectae  AH, 
BC,  quae  cum  incidente  AB  duos  ad  easdem  partes  an- 
gulos  efliciant  duobus  rectis  aequales,  ut  puta  utrunque 
rectum,  non  etiam  ipsae,  ad  eas  partes  in  infinitum  pro- 
tractae,  semper  magis  invicem  dissihant  ultra  omnem 
finitam  assignabilem  distantiam.  Quatenus  enim  partem 
afiirmativam  praesumere  quis  velit;  quae  utique  veris- 
sima  est  in  hypothesi  anguH  acuti;  non  erit  sane  legiti- 
mum  consequens,  quod  recta  AD  quomodoHbet  secans 
angulum  HAB;  unde  nempe  minores  fiant  duobus  rectis 
duo  simul  ad  easdem  partes  interni  anguH  DAB,  CBA; 
quod,  inquam,  ea  recta  AD,  in  infinitum  producta  coire 
tandem  debeat  cum  producta  BC;  etiamsi  aHas  demon- 


may  not  the  same  be  said  of  the  two  assumed  straight 
lines  AH,  AD,  unless  a  special  reason  constrains  to  the 
contrary  ? 

Nor  here  can  Clavius  be  blamed  that  he  opposes  to 
Proclus  this  property  of  the  Conchoid,  which  cannot  be 
demonstrated  except  with  the  aid  of  many  theorems  rest- 
ing  upon  the  here  controverted  postulate. 

For  I  say  from  this  itself  the  force  of  the  Clavian 
rebuttal  is  confirmed;  because  it  is  certain,  this  postulate 
being  assumed,  it  manifestly  follows,  that  two  Hnes  in 
infinitttm  protracted,  one  straight  and  the  other  curved, 
can  recede  one  from  the  other  ever  more  within  a  certain 
finite  determinate  Hmit;  whence  at  any  rate  may  arise  a 
suspicion  lest  the  same  may  happen  for  two  straight 
Hnes,  unless  otherwise  demonstrated. 

But  it  is  not  therefore  possible,  when  I  now  have  made 
manifest  in  the  corollary  to  the  preceding  [31]  proposition 
the  absolute  truth  of  the  aforesaid  assumption,  imme- 
diately  to  go  over  to  the  assertion  of  the  Euclidean  pos- 
tulate. 

For  previously  must  also  be  demonstrated,  that  those 
two  straights  AH,  BC,  which  with  the  transversal  AB 
make  two  angles  toward  the  same  parts  equal  to  two 
right  angles,  as  for  example  each  a  right  angle,  do  not 
also,  protracted  toward  these  parts  in  infinituni,  always 
separate  more  from  one  another  beyond  all  finite  assign- 
able  distance. 

For  if  one  chooses  to  presume  the  affirmative,  which 
is  indeed  entirely  true  in  the  hypothesis  of  acute  angle; 
it  certainly  will  not  be  a  legitimate  consequence,  that  a 
straight  AD  in  any  way  cutting  the  angle  HAB,  hence 
of  course  making  at  the  same  time  two  internal  angles 
DAB,  CBA  toward  the  same  parts  less  than  two  right 
angles;  that,  I  say,  this  straight  AD,  produced  in  infini- 
tiim,  must  at  length  meet  with  BC  produced;  even  if  it 

8s 


stratum  sit,  quod  distantia  duarum  AH,  AD  in  infinitum 
productarum  major  semper  evadat  ultra  omnem  finitum 
designabilem  limitem. 

Quod  autem  praefatus  Clavius  satis  esse  judicaverit 
veritatem  illius  assumpti  ad  demonstrandum  Pronuncia- 
tum  hic  controversum ;  condohari  id  debet  praeconceptae 
ab  ipso  Clavio  opinioni  circa  rectas  lineas  aequidistantes, 
de  quibus  in  sequente  Scholio  commodius  agemus. 

SCHOLION  II. 

In  quo  expenditur  idea  Clarissimi  Viri  Joannis  Alphonsi 
Borelli  in  suo  EucHde  Restituto. 

Accusat  doctissimus  hic  Auctor  EucHdem,  quod  rectas 
lineas  parallelas  eas  esse  definiverit,  quae  in  eodem  plano 
existentes  non  concurrunt  ad  utrasque  partes,  licet  in 
infinitum  producantur.  Rationem  accusationis  affert, 
quod  talis  [^2]  passio  ignota  sit:  tmn  quia,  inquit,  igno- 
ramus,  an  tales  lineae  infinitae  non  concurrentes  reperiri 
possint  in  natura:  tum  etiam  quia  infiniti  proprietates 
percipere  non  possumus,  et  proinde  non  est  evidenter 
cognita  passio  ejusmodi. 

Sed  pace  tanti  Viri  dictum  sit:  Numquid  reprehendi 
potest  Euclides,  quod  quadratum  (ut  unum  inter  innu- 
mera  exemplum  proferam)  definiverit  esse  figuram  qua- 
drilateram,  aequilateram,  rectangulam ;  cum  dubitari  pos- 
sit,  an  figura  ejusmodi  locum  habeat  in  natura?  Repre- 
hendi,  inquam,  aequissime  posset;  si,  ante  omnem  Prob- 
lematicam  demonstrativam  constructionem,  figuram  prae- 
dictam  assumpsisset  tanquam  datam.  Hujus  autem  vitii 
immunem  esse  EucHdem  ex  eo  manifeste  Hquet,  quod 
nusquam  praesumit  quadratum  a  se  definitum,  nisi  post 
Prop.   46.   Libri  primi,   in  qua  problematice  docet,   ac 


were  at  another  time  demonstrated,  that  the  distance  of 
the  two  AH,  AD  produced  in  infinitum  goes  out  ever 
greater  beyond  all  finite  assignable  Hmit. 

But  that  the  aforesaid  Clavius  should  have  judged 
the  truth  of  this  assumption  sufficient  for  demonstrating 
the  postulate  here  in  question;  that  ought  to  be  blamed 
to  the  opinion  preconceived  by  Clavius  about  equidistant 
straight  Hnes,  which  we  may  discuss  more  conveniently 
in  a  subsequent  schoHon. 

SCHOLION  II. 

In  which  is  weighed  an  idea  of  that  hrilliant  man  Giovanni 
Alfonso  Borelli  in  his  Euclides  Restitutus. 

This  most  learned  author  blames  EucHd,  because  he 
defines  paraHel  straight  Hnes  to  be  those,  which  being 
in  the  same  plane  do  not  meet  on  either  side,  even  if 
produced  in  infinitum. 

He  offers  as  ground  for  his  accusation,  that  such 
relation  [^2]  is  unknown:  first,  he  says,  because  we  are 
ignorant,  whether  such  infinite  non-concurrent  lines  can 
be  found  in  nature :  then  also  because  we  cannot  perceive 
the  properties  of  the  infinite,  and  hence  a  relation  of  this 
sort  is  not  clearly  cognized. 

But  with  reverence  f or  so  great  a  man  it  may  be  said ; 
Can  EucHd  be  blamed,  because  (to  bring  forward  one 
among  innumerable  examples)  he  defines  a  square  to  be  a 
figure  quadrilateral,  equilateral,  rectangular ;  when  it  may 
be  doubted,  whether  a  figure  of  this  sort  has  place  in 
nature?  He  could,  say  I,  most  justly  have  been  blamed. 
if,  before  as  a  problem  demonstrating  the  construction,  he 
had  assumed  the  aforesaid  figure  as  given. 

But  that  EucHd  is  f ree  from  this  fault  foHows  mani- 
festly  f rom  this,  that  he  nowhere  assumes  the  square  de- 
fined  by  him,  except  after  Prop.  46  of  the  First  Book, 

«7 


demonstrat  quadrati  prout  ab  ipso  definiti,  a  data  recta 
linea  descriptionem.  Simili  igitur  modo  reprehendi  ne- 
quit  Euclides,  quod  rectas  lineas  parallelas  eo  tali  modo 
definiverit,  cum  eas  nusquam  ad  constructionem  ullius 
Problematis  assumat  tanquam  datas,  nisi  post  Prop.  31. 
lib.  primi,  in  qua  Problematice  demonstrat,  quo  pacto 
a  dato  extra  datam  rectam  lineam  puncto  duci  debeat 
recta  linea  eidem  parallela,  et  quidem  juxta  definitionem 
ab  eo  traditam  parallelarum,  ita  ut  nempe  in  infinitum 
protractae  in  neutram  partem  sibi  invicem  occurrant: 
Quodque  amplius  est;  id  ipsum  demonstrat  sine  ulla  de- 
pendentia  a  Pronunciato  hic  controverso.  Itaque  Eucli- 
des  sine  ulla  petitione  principii  demonstrat  reperiri  posse 
in  natura  duas  tales  lineas  rectas,  quae  (in  eodem  plano 
consistentes )  in  utramque  partem  in  infinitum  protractae 
nunquam  concurrant;  ac  propterea  cognitam  nobis  evi- 
denter  facit  eam  passionem,  per  quam  rectas  Hneas  paral- 
lelas  definit. 

Pergamus  porro,  quo  nos  invitat  diHgens  EucHdis  ac- 
cusator.  ParaHelas  rectas  Hneas  appehat  duas  quasHbet 
[33]  rectas  AC,  BD,  quae  perpendiculariter  ad  easdem 
partes  (fig.  apud  me  21.)  insistant  uni  cuidam  rectae  AB. 
Nihil  moror,  quin  definitio  ejusmodi  exposita  sit  per  pas- 
sionem  (ut  ipse  ait)  possibilem,  et  evidentissimam ;  cum 
(ex  undecima  primi)  a  quoHbet  in  data  recta  puncto  ex- 
citari  possit  perpendicularis. 

Verum  hanc  ipsam  et  possibiHtatem,  et  evidentiam 
jam  demonstravi  circa  definitionem  traditam  ab  EucHde. 
Quare  unice  restat,  ut  conferatur  notum  ihud  Pronuncia- 
tum  EucHdaeum  cum  altero  itidem  Pronunciato,  quod 


88 


in  which  in  form  of  a  problem  he  teaches,  and  demon- 
strates  the  description  from  a  given  straight  line,  of  the 
square  as  defined  by  him. 

In  the  same  way  therefore  Euchd  ought  not  to  be 
blamed,  because  he  defined  parallel  straight  lines  in  this 
manner,  since  he  nowhere  assumes  them  as  given  for  the 
construction  of  any  problem,  except  after  Prop.  31  of  the 
First  Book,  in  which  as  a  problem  he  demonstrates,  how 
shoidd  be  drawnfrom  agiven  point  without  a  given  straight 
line  a  straight  line  parallel  to  this,  and  indeed  according  to 
the  definition  of  parallels  given  by  him,  so  that  produced 
indeed  into  the  infinite  on  neither  side  do  they  meet  one 
another.  And  what  is  more;  he  demonstrates  this  with- 
out  any  dependence  from  the  postulate  here  controverted. 
Thus  EucHd  demonstrates  without  any  petitio  principii 
that  there  can  be  found  in  nature  two  such  straight  lines, 
which  (lying  in  the  same  plane)  protracted  on  each  side 
into  the  infinite  never  meet,  and  therefore  makes  clearly 
known  to  iis  that  relation  by  which  he  defines  parallel 
straight  Hnes. 

Let  us  continue  onward,  whither  the  scrupulous  ac- 
cuser  of  EucHd  invites  us.  ParaHel  straight  Hnes  he  calls 
any  two[33]straights  AC,  BD,  which 
toward  the  same  parts  stand  at  right 
angles  to  a  certain  straight  AB  (fig. 
with  me  21).  I  admit  that  such  a 
definition  is  set  forth  by  a  state 
(as  he  says)  possible  and  most  evi- 
dent;  since  (Eu.  I.  11)  from  any 
point  in  the  given  straight  a  perpendicular  can  be  erected. 

But  precisely  both  this  possibiHty  and  clearness  I  have 
just  now  demonstrated  about  the  definition  propounded 
by  EucHd. 

Wherefore  remains  only  to  compare  that  known  pos- 
tulate  of  EucHd  with  the  other  Hke  postulate,   which 

89 


usui  esse  debeat  ad  ulteriorem  progressum  post  novam 
istam  parallelarum  definitionem.  Ecce  autem  alterum 
istud  Pronunciatum  apud  Clavium  (ad  quem  diserte  pro- 
vocat  ipse  Borellius)  in  Scholio  post  Prop.  28.  lib.  primi: 
Si  recta  linea,  ut  puta  BD  super  aliam  rectam,  ut  puta 
BA,  in  transversum  moveatur  constituens  cum  ea  in  suo 
extremo  B  angulos  semper  rectos,  describet  alterum  illius 
extremum  D  lineam  quoque  rectam  DC,  dum  nempe  ipsa 
BD  pervenerit  ad  congruendum  alteri  aequali  AC. 

Agnosco  opportunitatem  Pronunciati,  ut  inde  transi- 
tus  fiat  ad  demonstrandum  illud  alterum  Euclidaeum,  quo 
nempe  fulciri  tandem  debet  reliqua  omnis  Geometria. 
Nam  antea  proposuerat  Clavius ;  quod  linea,  cujus  omnia 
puncta  aeque  distent  a  quadam  supposita  recta  AB ;  qualis 
utique  est  (ex  hypothesi  praedictae  descriptionis)  linea 
DC;  debet  esse  etiam  ipsa  linea  recta;  quia  nempe  ejus- 
modi  erit,  ut  omnia  ipsius  puncta  intermedia  ex  aequo 
jaceant  (quahs  est  rectae  hneae  definitio)  inter  ejus  ex- 
trema  puncta  D,  et  C;  ex  aequo,  inquam,  jaceant;  cum 
omnia  aeque  distent  ab  ea  supposita  recta  AB,  nimirum 
quanta  est  longitudo  ipsius  BD,  aut  AC.  Quo  loco  affert 
Clavius  exemplum  lineae  circularis,  de  qua  commodius 
inf ra  disseremus ;  ubi  ostendam  clarissimam  hac  in  parte 
disparita-[34]tem  inter  Hneam  rectam,  et  circularem.  Nam 
interim  dico  non  satis  liquere,  an  linea  descripta  ab  eo 
puncto  D  sit  potius  recta  DC,  quam  curva  quaedam  DGC 
seu  convexa,  seu  concava  versus  partes  ipsius  BA. 

Si  enim  ex  puncto  F  dividente  bifariam  ipsam  BA 
intelligatur  educta  perpendicularis,  quae  occurrat  rectae 
DC  in  E,  et  praedictis  curvis  in  G,  et  G,  constat  sane 
(ex  2.  hujus)  rectos  fore  angulos  hinc  inde  ad  punctum 


90 


must  be  used  for  farther  progress  after  the  new  definition 
of  parallels. 

But  behold  this  other  postulate  in  Clavius  (to  whom 
BorelH  himself  expressly  refers)  in  the  schoHon  after 
Prop.  28  of  the  First  Book:  If  a  straight  Hne,  as  sup- 
pose  BD  upon  another  straight,  as  suppose  BA,  moves 
transversely  making  with  it  at  its  extremity  B  always 
right  angles,  its  other  extremity  D  describes  a  Hne  also 
straight  DC,  until  this  BD  shaH  have  come  to  congruence 
with  the  other  equal  sect  AC.  I  acknowledge  the  fitness  of 
the  postulate,  that  thence  a  transit  may  be  made  to  demon- 
strating  that  other  EucHdean  postulate,  upon  which  cer- 
tainly  at  length  must  be  supported  aU  remaining  geom- 
etry.  For  Clavius  had  previously  declared ;  that  a  line,  of 
which  all  points  are  equally  distant  from  a  certain  as- 
sumed  straight  AB ;  as  assuredly  is  ( f rom  the  hypothesis 
of  the  aforesaid  construction)  the  line  DC;  this  line  also 
must  be  straight ;  because  certainly  it  will  be  of  such  sort, 
that  all  its  intermediate  points  lie  ex  aeqno  (such  is  the 
definition  of  a  straight  line)  between  its  extreme  j)oints 
D,  and  C ;  lie  ex  aeqiio,  say  I,  since  all  are  equally  distant 
from  this  assumed  straight  AB,  truly  by  as  much  as  the 
length  is  of  this  BD,  or  AC.  In  this  place  Clavius  intro- 
duces  the  example  of  the  circular  line,  of  which  we  shall 
speak  more  conveniently  below;  where  I  shall  show  the 
clearest  disparity  in  this  regard  [^4]  between  the  straight 
line  and  circle. 

But  meanwhile  I  say  it  is  not  sufficiently  evident, 
whether  the  line  described  by  this  point  D  is  rather  the 
straight  DC  than  a  certain  curve  DGC  either  convex  or 
concave  toward  the  side  of  this  BA. 

For  if  from  the  point  F  bisecting  this  BA  a  perpen- 
dicular  is  supposed  erected,  which  meets  the  straight  DC 
in  E,  and  the  aforesaid  curves  in  G,  and  G,  it  follows 
surely  (from  P.  II.)  that  the  angles  at  the  point  E  will 

91 


E;  qualiscunque  tandem  in  eo  motu  intelligatur  descripta 
linea  DC  a  puncto  D;  ac  praeterea  (ex  facili  intellecta 
superpositione)  aequales  hinc  inde  fore  angulos  ad  puncta 
G,  prout  alterutra  curva  DGC  descripta  fuerit. 

Sed  rursum;  assumpto  in  AB  quolibet  puncto  M;  si 
educatur  perpendicularis,  quae  occurrat  rectae  DC  in  N, 
et  praedictis  curvis  in  H,  et  H,  paulo  post  demonstrabo 
rectos  fore  angulos  hinc  inde  ad  punctum  N,  quatenus 
quidem  recta  ipsa  DC  genita  supponatur  in  suo  illo  motu 
a  puncto  D,  seu  quatenus  recta  MN  aequaHs  censeatur 
ipsi  BD.  Sin  vero  alterutra  curva  DHC  genita  putetur; 
ex  facili  itidem  praescripta  superpositione  demonstrabi- 
tur  aequales  rursum  hinc  inde  fore  angulos  MHD,  MHC, 
ubivis  in  ea  alterutra  descripta  curva  sumptum  fuerit 
punctum  H,  ex  quo  ad  subjectam  rectam  Hneam  AB  de- 
missa  inteHigatur  perpendicularis  HM.  Verum  hac  de 
re  fusius,  ac  diHgentius  in  altera  parte  hujus  Hbri,  ubi 
locum  proprium  habet. 

Quorsum  igitur,  inquies,  praecox  ista  anticipatio  ?  In 
eum,  inquam,  finem ;  ut  ne  ex  ista  Hneae  eo  modo  genitae 
verissima,  et  a  me  exactissime  in  praecitato  loco  demon- 
stranda  proprietate;  et  quidem  citra  omnem  defectum 
quomodoHbet  infinite  parvum;  praecipitanter  censeremus 
non  nisi  rectam  Hneam  esse  posse.  SciHcet  hic  inquiritur 
penitior  rectae  Hneae  natura,  sine  qua  vix  infantiam  prae- 
[35]tergressa  Geometria  subsistere  ibi  deberet.  Non  igi- 
tur  hac  in  re  vituperari  potest  major  quaedam  exactis- 
simae  veritatis  inquisitio. 

Neque  tamen  hic  renuo,  quin  diHgentissima  aHqua 
experientia  physica  deprehendi  possit,  quod  Hnea  DC  eo 


92 


be  right,  whatever  line  DC  is  understood  at  length  as 
described  in  this  motion  by  the  point  D;  and  moreover 
(from  an  easily  understood  superposition)  the  angles  at 
the  points  G  v^ill  be  equal  according  as  the  one  or  the 
other  curve  DGC  may  be  described. 

But  again;  any  point  M  in  AB  being  assumed;  if  a 
perpendicular  is  erected,  which  meets  the  straight  DC  in 
N,  and  the  aforesaid  curves  in  H  and  H,  I  shall  prove  a 
little  later  that  the  angles  on  both  sides  at  the  point  N 
will  be  right,  in  so  far  indeed  as  this  straight  DC  is  sup- 
posed  generated  by  the  point  D  in  that  motion  of  its,  or 
in  as  far  as  the  straight  MN  is  decided  equal  to  this  BD. 

But  if  one  or  the  other  curve  DHC  is  supposed  gen- 
erated;  from  the  like  aforesaid  easy  superposition  will 
be  demonstrated  that  again  the  angles  MHD,  MHC  on 
both  sides  will  be  equal,  wherever  in  the  one  or  the  other 
described  curve  the  point  H  may  be  assumed,  from 
which  to  the  underlying  straight  line  AB  the  perpendicu- 
lar  HM  is  understood  as  let  fall.  But  of  this  thing  more 
fully  and  more  scrupulously  in  the  Second  Part  of  this 
Book,  where  it  has  its  proper  place. 

To  what  end  therefore,  will  you  say,  this  untimely 
anticipation  ? 

To  this  end,  say  I ;  lest  from  this  indubitable  property 
of  the  line  generated  in  this  manner,  proved  by  me  most 
rigorously  in  the  aforesaid  place ;  and  indeed  beyond  any 
defect  of  any  sort  infinitely  small;  we  may  decide  pre- 
cipitately  that  the  line  can  be  only  the  straight. 

Obviously  the  nature  of  the  straight  line  must  here  be 
investigated  more  profoundly,  without  which  geometry 
scarcely  grown  beyond  infancy  [^5]  must  there  remain. 
Therefore  in  this  affair  cannot  be  blamed  a  certain  greater 
investigation  of  a  most  exact  verity. 

Nor  yet  do  I  here  deny,  but  that  by  some  most  ac- 
curate  physical  experimentation  may  be  discovered,  that 

93 


motu  genita  non  nisi  recta  linea  censenda  sit.  Sed  quate- 
nus  ad  experientiam  physicam  provocare  hic  Hceat;  tres 
statim  afferam  demonstrationes  Physico-Geometricas  ad 
comprobandum  Pronunciatum  EucHdaeum.  Ubi  non  lo- 
quor  de  experientia  physica  tendente  in  infinitum,  ac 
propterea  nobis  impossibiH ;  quaHs  nempe  requireretur  ad 
cognoscendum,  quod  puncta  omnia  junctae  rectae  DC 
aequidistent  a  recta  AB,  quae  supponitur  in  eodem  cum 
ipsa  DC  plano  consistens.  Nam  mihi  satis  erit  unicus 
individuus  casus;  ut  puta,  si  juncta  recta  DC,  assump- 
toque  uno  aHquo  ejus  puncto  N,  perpendicularis  NM 
demissa  ad  subjectam  AB  comperiatur  esse  aequaHs  ipsi 
BD,  sive  AC.  Tunc  enim  anguH  hinc  inde  ad  punctum 
N  aequales  forent  (ex  1.  hujus)  anguHs  sibi  correspon- 
dentibus  ad  puncta  C,  et  D,  qui  rursum  (ex  eadem  1. 
hujus)  aequales  inter  se  forent.  Quare  anguH  hinc  inde 
ad  punctum  N,  atque  ideo  etiam  reHqui  duo  recti  erunt. 
Igitur  unum  habebimus  casum  pro  hypothesi  anguH  recti ; 
ac  propterea  (juxta  quintam,  et  decimamtertiam  hujus) 
demonstratum  habebimus  Pronunciatum  EucHdaeum.  At- 
que  haec  esse  potest  prima  demonstratio  Physico-Geo- 
metrica. 

Transeo  ad  secundam.  Esto  semicirculus,  cujus  cen- 
trum  D,  et  diameter  AC.  Si  ergo  (fig.  17.)  in  ejus  cir- 
cumferentia  assumatur  punctum  aliquod  B,  ad  quod  junc- 
tae  AB,  CB  comperiantur  continere  angulum  rectum,  sa- 
tis  erit  hic  unicus  casus  (prout  demonstravi  in  18.  hujus) 
ad  stabiliendam  hypothesim  anguli  recti,  ac  propterea  (ex 
praedicta  13.  hujus)  ad  demonstrandum  notum  illud  Pro- 
nunciatum.  [36] 

Superest    tertia    demonstratio    Physico-Geometrica, 


94 


the  line  DC  generated  by  this  motion  can  only  be  adjudged 
a  straight  line. 

But  in  so  far  as  may  be  here  permissible  to  cite  phys- 
ical  experimentation,  I  forthwith  bring  forward  three 
demonstrations  physico-geometric  to  sanction  the  EucHd- 
ean  postulate. 

Therewith  I  do  not  speak  of  physical  experimentation 
extending  into  the  infinite,  and  therefore  impossible  for 
us ;  such  as  of  course  would  be  requisite  to  the  cognizing, 
that  all  points  of  the  straight  join  DC  are  equidistant 
from  the  straight  AB,  which  is  supposed  to  be  in  the  same 
plane  with  this  DC. 

For  a  single  individual  case  will  be  sufficient  f or  me ; 
as  suppose,  if,  the  straight  DC  being  joined,  and  any  one 
point  of  it  N  being  assumed,  the  perpendicular  NM  let 
fall  to  the  underlying  AB  is  ascertained  to  be  equal  to 
BD  or  AC.  For  then  the  angles  on  both  sides  at  the 
point  N  would  be  equal  (P.  I.)  to  the  angles  correspond- 
ing  to  them  at  the  points  C  and  D,  which  again  (from 
the  same  P.  I.)  would  be  equal  inter  se.  Wherefore  the 
angles  on  both  sides  at  the  point  N,  and  therefore  also 
the  remaining  two  will  be  right. 

Therefore  we  shall  have  a  case  for  the  hypothesis  of 
right  angle;  and  consequently  (by  Propp.  V.  and  XIII.) 
we  shall  have  demonstrated  the  Euclidean  postulate.  And 
this  may  be  the  first  demonstration  physico-geometric. 

I  pass  over  to  the  second.  Let  there  be  a  semi-circle, 
of  which  the  center  is  D,  and  diameter  AC.  If  then 
(fig.  17)  any  point  B  is  assumed  in  its  circumference,  to 
which  AB,  CB  joined  are  ascertained  to  contain  a  right 
angle,  this  single  case  will  be  sufficient  (as  I  have  demon- 
strated  in  P.  XVIII.)  for  establishing  the  hypothesis  of 
right  angle,  and  consequently  (from  the  aforesaid  P. 
XIII. )  f or  demonstrating  that  f amous  postulate.  i^] 

There  remains  the  third  demonstration  physico-geo- 

95 


quam  puto  omnium  efficacissimam,  ac  simplicissimam, 
utpote  quae  subest  communi,  facillimae,  paratissimaeque 
experientiae.  Si  enim  in  circulo,  cujus  centrum  D,  tres 
coaptentur  (fig.  22.)  rectae  lineae  CB,  BL,  LA,  aequales 
singulae  radio  DC,  comperiaturque  juncta  AC  transire 
per  centrum  D,  satis  id  erit  ad  demonstrandum  intentum, 
Nam  junctis  DB,  DL,  tria  habebimus  triangula,  quae 
(ex  8.  et  5.  primi)  tum  inter  se  invicem,  tum  etiam  in  se 
ipsis  singula  erunt  aequiangula.  Quoniam  igitur  tres 
simul  anguli  ad  punctum  D,  nimirum  ADL,  LDB,  BDC 
aequales  sunt  (ex  13.  primi)  duobus  rectis;  duobus  etiam 
rectis  aequales  erunt  tres  simul  anguli  cujusvis  illorum 
triangulorum,  ut  puta  trianguli  BDC.  Quare  (ex  15. 
hujus)  stabilita  hinc  erit  hypothesis  anguli  recti;  ac  prop- 
terea  (ex  jam  nota  13.  hujus)  demonstratum  manebit 
illud  Pronunciatum. 

Sin  vero,  ante  omnem  attentatam  seu  demonstratio- 
nem,  seu  figuralem  exhibitionem,  conferre  inter  se  pla- 
ceat  duo  illa  Pronunciata,  fateor  sane  Euclidaeum  videri 
posse  obscurius,  aut  etiam  falsitati  obnoxium.  At  post 
figuralem  exhibitionem,  quam  Scholio  IV.  consequenti  re- 
servo,  constabit  viceversa  Pronunciatum  quidem  Eucli- 
daeum  retinere  posse  dignitatem,  ac  nomen  Pronunciati, 
alterum  vero  inter  Theoremata  computari  tutius  debere. 

Sed  hic  explicare  debeo  (prout  paulo  ante  me  factu- 
rum  spopondi)  manifestam  isto  in  genere  disparitatem 
inter  lineam  circularem,  et  lineam  rectam.  Disparitas 
autem  ex  eo  oritur ;  quod  recta  quidem  linea  dicitur  ad  se 
ipsam ;  circularis  vero,  ut  puta  (fig.  23.)  MDHNM,  non  ad 


96 


metric,  which  I  think  the  most  efficacious  and  most  simple 
of  all,  inasmuch  as  it  rests  upon  an  accessible,  most  easy, 
and  most  convenient  experiment. 

For  if  in  a  circle,  whose  center  is 
D,  are  fitted  (fig.  22)  three  straight 
lines  CB,  BL,  LA,  each  equal  to  the 
radius  DC,  and  it  is  ascertained  that 
the  join  AC  goes  through  the  center 
D,  this  will  be  sufficient  for  demonstrating  the  assertion. 

For,  DB,  DL  being  joined,  we  will  have  three  tri- 
angles,  which  (from  Eu.  L  8  and  5)  not  only  will  be 
equiangular  to  one  another,  but  also  singly  for  themselves. 
Therefore  since  the  three  angles  together  at  the  point  D, 
indeed  ADL,  LDB,  BDC  are  equal  (by  Eu.  I.  13)  to 
two  right  angles;  also  the  three  angles  together  of  each 
of  these  triangles  will  be  equal  to  two  right  angles,  as 
suppose  of  the  triangle  BDC.  Wherefore  (from  P.  XV.) 
will  be  established  hence  the  hypothesis  of  right  angle; 
and  consequently  (from  the  already  admitted  P.  XIIL) 
that  postulate  will  be  demonstrated. 

But  if,  before  all  attempt  whether  at  demonstration 
or  at  graphic  exhibition,  one  wishes  to  compare  infer  se 
those  two  postulates,  I  grant  indeed  the  Euclidean  may 
appear  more  obscure  or  even  liable  to  objection.  But 
after  the  graphic  exhibition  which  I  reserve  for  Schblion 
IV  following,  it  will  appear  vice  versa  that  the  Euclidean 
postulate  indeed  can  retain  the  dignity  and  name  of  postu- 
late,  but  the  other  ought  rather  to  be  reckoned  among  the 
theorems. 

But  here  I  must  explain  (as  a  little  above  I  have  prom- 
ised  I  was  about  to  do)  the  manifest  disparity  in  this 
relation  between  the  circular  line  and  the  straight  line. 
Now  the  disparity  arises  from  this;  that  a  line  is  called 
straight  in  reference  to  itself;  but  is  called  circular,  as 
suppose  (fig.  23)  MDHNM,  not  in  reference  to  itself, 

97 


se  ipsam,  sed  ad  alterum  dicitur,  nimirum  ad  quoddam  al- 
terum  in  eodem  cum  ipsa  plano  existens  punctum  A,  quod 
est  ejusdem  centrum.     Consequens  igitur  est,  prout  opti- 
[37]me   demonstratur   a   Clavio,    quod    linea    FBCL   in 
eodem  cum  illa  plano  consistens,  et  cujus  omnia  puncta 
aequidistent  a  praedicta  MDHNM,  sit  et  ipsa  circularis, 
nimirum  omnibus  suis  punctis  aequidistans  a  communi 
centro  A.    Quod  enim  BD,  quae  sit  continuatio  in  rectum 
ipsius  AB,  sit  mensura  distantiae  illius  puncti  B  ab  ea 
circulari  MDHNM,  ex  eo  constat;  quia  (ex  7.  tertii,  quae 
est  independens  a  Pronunciato  hic  controverso)  minima 
omniuni  ipsa  est,  quae  ab  eo  puncto  in  eam  circumferen- 
tiam  cadere  possint.  Idem  valet  de  reliquis  CH,  LN,  FM. 
Quoniam  igitur  et  totae  AM,  AD,  AH  aequales  sunt, 
utpote  radii  ex  centro  A  ad  suppositam  lineam  circularem. 
MDHNM;  atque  item  aequales  sunt  abscissae  FM,  BD, 
CH,  LN,  quae  nempe  mensura  sunt  aequalis  distantiae 
omnium  punctorum  illius  lineae  FBCLF  ab  ea  supposita 
linea  circulari  MDHNM ;  consequens  plane  est,  ut  aequa- 
les  pariter  sint  residuae  AF,  AB,  AC,  AL,  ac  propterea 
ipsa  etiam  linea  FBCLF  sub  eodem  centro  A  circularis  sit. 
Numquid  autem  uniformiter,  ad  demonstrandum,  quod 
linea  DC  (fig.  2L)  eo  tali  motu  genita  a  puncto  D  sit 
linea  recta,  satis  erit  aequidistantia  omnium  ipsius  punc- 
torum  a  subjecta  recta  AB?     Nullo  modo.     Nam  linea 
recta  dicitur  absolute  ad  se  ipsam,  sive  in  se  ipsa,  nimi- 
rum  ita  ex  aequo  jacens  inter  siia  puncta,  ac  praesertim 
extrema,  ut  manentibus  istis  immotis  nequeat  ipsa  revolvi 
ad  occupandum  novum  locum.     Nisi  haec  passio  aliquo 
pacto  demonstretur  de  ea  DC,  nunquam  constabit  eam 


but  to  something  else,  f orsooth  to  a  certain  other  point 
A  existing  in  the  same  plane  with  it,  which  is  its  center. 

The  consequence  therefore  is,  as  is 
most  excellently  1^7]  demonstrated  by 
Clavius,  that  the  line  FBCL  existing  in 
the  same  plane  with  it,  and  whose 
points  are  all  equidistant  from  the 
aforesaid  MDHNM,  is  also  itself  cir-  ^^s-  23- 

cular,  truly  equidistant  in  all  its  points  from  the  common 
center  A.  That  in  fact  BD,  which  is  the  continuation  in 
a  straight  of  AB,  is  the  measure  of  the  distance  of  that 
point  B  from  this  circle  MDHNM  follows  from  this; 
because  (from  Eu.  ni.  7,  which  is  independent  of  the 
postulate  here  in  controversy)  this  is  the  smallest  of  all, 
which  can  fall  from  this  point  upon  this  circumference. 
The  same  holds  of  the  remaining  CH,  LN,  FM. 

Since  therefore  also  the  wholes  AM,  AD,  AH,  are 
equal  as  radii  from  the  center  A  to  the  line  assumed 
circular  MDHNM;  and  also  the  sections  FM,  BD,  CH, 
LN  are  equal,  which  obviously  are  the  measure  of  the 
equal  distance  of  all  points  of  that  line  FBCLF  from  this 
line  presumed  circular  MDHNM ;  the  consequence  plainly 
is,  that  equal  likewise  are  the  remainders  AF,  AB,  AC, 
AL,  and  therefore  also  this  line  FBCLF  is  a  circle  with 
the  same  center  A. 

But  now  likewise,  for  demonstrating  that  the  line 
DC  (fig.  21)  generated  through  such  a  motion  by  the 
point  D  is  a  straight  line  will  the  equidistance  of  all  its 
points  from  the  underlying  straight  AB  be  sufificient? 
In  no  way. 

For  a  line  is  called  straight  absolutely  in  reference  to 
itself,  or  in  itself,  doubtless  as  lying  ex  aeqiio  hetween  its 
pointSj  and  especially  end  points,  so  that  these  remaining 
unmoved  it  cannot  be  revolved  into  occupying  a  new  place. 
Unless  this  state  in  some  way  be  demonstrated  of  this 

99 


esse  lineam  rectam,  qualiscunque  tandem  supponatur,  aut 
demonstretur  omnium  ipsius  punctorum  relatio  ad  sub- 
jectam  in  eodem  plano  rectam  AB;  praesertim  vero,  ne 
uniformiter  dicamus  nullam  aliam  in  eo  plano  fore  lineam 
rectam,  quae  omnibus  suis  punctis  non  aequidistet  ab  ea 
supposita  recta  linea  AB.  [38] 

Neque  tamen  dictum  hoc  meum  ita  accipi  volo,  quasi 
putem  demonstrari  non  posse,  quod  linea  sic  genita  ipsa 
sit  linea  recta,  nisi  post  demonstratam  veritatem  contro- 
versi  Pronunciati ;  cum  magis  ego  ipse  prope  finem  hujus 
Libri  demonstraturus  id  sim,  ad  confirmandum  ipsum  tale 
Pronunciatum. 

SCHOLION  III. 

In  quo  expenditur  conatus  Nassaradini  Arahis,  et  simul 

idea  super  eodem  negotio  Clariss.  Viri 

Joannis  Vallisii. 

Conatum  istum  Nassaradini  Arabis  latino  idiomate 
typis  vulgavit  praelaudatus  Vir  Joannes  ValHsius,  cum 
animadversionibus  opportuno  loco  adjectis.  Duo  autem 
in  rem  suam  postulat  sibi  concedi  Nassaradinus. 

Primum  est ;  ut  duae  quaeHbet  rectae  Hneae  in  eodem 
plano  positae,  in  quas  aHae  quotHbet  rectae  Hneae  ita  in- 
cidant,  ut  uni  quidem  earum  perpendiculares  semper  sint, 
alteram  vero  ad  angulos  inaequales  semper  secent,  nimi- 
rum  versus  unam  partium  sub  angulo  semper  acuto,  et 
versus  alteram  sub  angulo  semper  obtuso;  ut,  inquam, 
priore  loco  dictae  Hneae  censeantur  semper  magis  (quan- 
diu  se  mutuo  non  secent)  ad  se  invicem  accedere  versus 
partes  iHorum  angulorum  acutorum;  et  vicissim  semper 
magis  a  se  invicem  recedere  versus  partes  angulorum 
obtusorum. 

At  ego  quidem,  si  nihil  aHud  moratur  Nassaradinum, 


DC  it  will  never  be  certaiii  that  this  is  a  straight  line, 
whatever  relation  finally  is  supposed  or  demonstrated  of 
all  its  points  to  the  underlying  straight  AB  in  the  same 
plane;  but  especially  we  must  not  say  analogically  that 
no  other  Hne  in  this  plane  will  be  straight  which  in  all 
its  points  is  not  equidistant  from  this  Hne  AB  supposed 
straight.  [38] 

Nor  finally  do  I  wish  this  dictum  of  mine  so  taken, 
as  if  I  think  it  cannot  be  demonstrated,  that  the  Hne  thus 
generated  is  itself  a  straight  Hne,  except  after  truth  dem- 
onstrated  of  the  controverted  postulate;  since  rather  I 
myself  wiH  demonstrate  it  toward  the  end  of  this  Book, 
for  confirming  such  postulate  itself. 

SCHOLION  III. 

Inwhich  isweighed  the  endeavor  of  theArab  Nasiraddin, 

and  likewise  the  idea  of  the  illustrious  John 

Wallis  upon  the  same  affair. 

This  endeavor  of  the  Arab  Nasiraddin  the  above 
eulogized  John  WaHis  has  pubHshed  in  the  Latin  language 
with  remarks  added  in  opportune  place. 

However  Nasiraddin  requires  two  things  to  be  con- 
ceded  to  him  in  this  afTair. 

The  first  is;  that  any  two  straight  Hnes  lying  in  the 
same  plane,  upon  which  ever  so  many  other  straight  Hnes 
50  strike,  that  they  are  always  perpendicular  to  one  indeed 
of  these,  but  always  cut  the  other  at  unequal  angles,  truly 
toward  one  part  always  under  an  acute  angle,  and  toward 
the  other  always  under  an  obtuse  angle;  that,  I  say,  the 
above-mentioned  Hnes  be  supposed  always  more  (as  long 
as  they  do  not  mutuahy  cut)  to  approach  each  other  toward 
the  side  of  those  acute  angles;  and  on  the  other  hand 
always  more  to  recede  f rom  one  another  toward  the  parts 
of  the  obtuse  angles. 

But  I  indeed,  if  nothing  else  impedes  Nasiraddin,  wil- 


libens  permitto,  quod  postulat;  cum  istud  ipsum,  quod 
ab  eo  indemonstratum  relinquitur,  intelligi  possit  exac- 
tissime  a  me  demonstratum  in  Cor.  II.  post  3.  hujus. 

Alterum  Nassaradini  Postulatum  est  reciprocum  pri- 
mi ;  ut  nempe  acutus  semper  sit  angulus  versus  eas  partes, 
[29]  ad  quas  jam  dictae  perpendiculares  supponantur  fieri 
semper  breviores;  obtusus  autem  versus  alias  partes,  ad 
quas  eaedem  perpendiculares  supponantur  evadere  semper 
longiores. 

Verum  hic  latet  aequivocatio.  Cur  enim  (dum  ab 
una  aHqua  statuta  tanquam  prima  perpendiculari  proce- 
datur  ad  ahas)  consequentium  perpendicularium  anguli, 
ad  eandem  partem  acuti,  non  fiant  semper  majores,  quo 
usque  incidatur  in  angulum  rectum,  nimirum  in  talem 
perpendicularem,  quae  ipsa  sit  utriusque  praedictarum 
rectarum  commune  perpendiculum  ?  Et  istud  quidem  si 
accidat,  evanescit  latebrosa  ista  Nassaradini  praeparatio, 
postquam  ingeniose  quidem,  sed  magno  cum  labore  Euch- 
daeum  Pronunciatum  demonstrat. 

Quod  si  Nassaradinus  jure  quodam  suo  praesumere 
veht  tanquam  per  se  notam  consistentiam  iUam  ad  eandem 
partem  angulorum  acutorum:  Cur  non  etiam  (dicam  cum 
Vahisio)  concipi  potest  tanquam  per  se  clarum :  Duas  rec- 
tas  in  eodem  plano  convergentes  (in  quas  nempe  aha  recta 
incidens  duos  ad  easdem  partes  angulos  efficiat  minores 
duobus  rectis,  ut  puta  unum  rectum,  et  alterum  quomo- 
dohbet  acutum)  tandem  occursuras,  si  producantur?  Ne- 
que  enim  opponi  potest,  quod  major  ista  ad  unas  partes 
convergentia  subsistere  semper  possit  intra  quendam  de- 
terminatum  hmitem,  adeo  ut  nempe  tanta  quaedam  dis- 
tantia  inter  eas  hneas  ad  eam  partem  semper  intersit, 
etiamsi  caeteroquin  una  ad  alteram  semper  propius  acce- 


lingly  permit  what  he  postulates;  since  just  that,  which 
with  him  remains  undemonstrated,  can  be  recognized  as 
most  rigorously  demonstrated  by  me  in  Cor.  II.  to  P.  III 

The  other  postulate  of  Nasiraddin  is  the  reciprocal 
of  the  first;  that  indeed  the  angle  may  always  be  acute 
toward  those  parts  [^9]  where  the  just  mentioned  per- 
pendiculars  are  supposed  to  become  shorter;  but  obtuse 
toward  the  other  parts  where  these  perpendiculars  are 
supposed  to  go  out  always  longer.  But  here  lurks  an 
ambiguity. 

For  why  (while  from  any  one  perpendicular  pre- 
scribed  as  the  first  we  proceed  to  the  others)  may  not  the 
angles  of  the  consequent  perpendiculars,  on  the  same  side 
acute,  not  become  ever  greater,  even  to  where  one  strikes 
upon  a  right  angle,  consequently  upon  such  a  perpendicu- 
lar  as  is  itself  the  common  perpendicular  to  each  of  the 
aforesaid  straights?  And  if  indeed  that  happens,  evan- 
ishes  this  subtle  preparation  of  Nasiraddin,  by  means  of 
which  ingeniously  indeed,  but  with  great  labor  he  demon- 
strates  the  Euclidean  postulate. 

And  yet  if  Nasiraddin  with  a  certain  justice  may  de- 
termine  to  presume  as  if  known  per  se  that  persistence 
of  acute  angles  on  the  same  side:  why  cannot  also  (I 
speak  with  Wallis)  be  assumed  as  if  clear  per  se:  Two 
straights  in  the  same  plane  converging  (upon  which  of 
course  another  straight  striking  makes  toward  the  same 
parts  two  angles  less  than  two  right  angles,  as  suppose 
one  right,  and  the  other  in  whatever  way  acute)  finally 
meet,  if  produced? 

Nor  in  fact  can  it  be  objected,  that  this  greater  con- 
vergence  toward  one  side  can  always  subsist  within  a 
certain  determinate  limit,  so  that  indeed  a  certain  so 
much  of  distance  always  intervenes  between  these  lines 
on  this  side,  even  if  still  one  approaches  always  more 
nearly  to  the  other. 

103 


dat.  Non,  inquam,  opponi  id  potest ;  quoniam  ex  hoc  ipso 
demonstrabo,  post  XXV.  hujus,  omnium  taHum  recta- 
rum  ad  finitam  distantiam  occursum,  juxta  Pronuncia- 
tum  Euclidaeum. 

Jam  transeo  ad  praelaudatum  Joannem  VaUisium, 
qui  nempe,  ut  morem  gereret  tot  Magnis  Viris,  Veteri- 
bus  pariter,  ac  Recentioribus,  et  rursum  ex  onere  Cathe- 
[40]drae  suae  Oxoniensi  imposito,  hoc  idem  pensum  ag- 
gredi  voluit  demonstrandi  saepe  dictum  Pronunciatum. 
Unice  autem  assumit  tanquam  certum,  quod  sequitur :  ni- 
mirum  Datae  cuicunque  figurae  similem  aliam  cujuscun- 
que  magnitudinis  possihilem  esse.  Et  id  quidem  praesumi 
posse  de  quaHbet  iigura  (etiam  si  in  rem  suam  unice 
assumat  triangularem  rectiHneam)  bene  argumentatur 
ex  circulo,  quem  sciHcet  sub  quantoHbet  radio  describi 
posse  omnes  agnoscunt.  Deinde  acutus  Vir  cautissime 
observat  praesumptioni  huic  suae  non  obstare,  quod  prae- 
ter  correspondentium  angulorum  aequaHtatem  requiratur 
etiam  correspondentium  omnium  laterum  proportionaH- 
tas,  ut  habeatur  una  figura  rectiHnea,  v.  g.  triangularis, 
alteri  rectiHneae  triangulari  simiHs;  cum  tamen  Propor- 
tionaHum,  ac  subinde  simiHum  Figurarum  definitio  ex 
Quinto,  ac  Sexto  EucHdis  Libro  desumendae  sint :  Poterat 
enim  Euclides  (inquit  ipse)  utramque  Libro  Primo  prae- 
misisse.  Porro  autem,  hoc  stante  (quod  tamen  negari  a 
quopiam  posset,  nisi  demonstretur)  intentum  suum  pul- 
chro  sane,  atque  ingenioso  moHmine  exequitur  laudatus 
Vir. 

Sed  nolo  oneri  a  me  suscepto  in  quoquam  deesse. 
Itaque  assumo  duo  triangula,  unum  ABC,  et  alterum 
DEF   (fig.  24.)    invicem  aequiangula:   Non  dico  plane 


104 


That  cannot,  I  say,  be  objected;  since  from  this  itself 
I  shall  demonstrate,  after  P.  XXV.,  the  meeting  at  a  finite 
distance  of  all  such  straights,  in  accordance  with  the 
EucHdean  postulate. 

Now  I  go  over  to  the  aforesaid  John  WalHs,  who,  as 
made  a  custom  with  so  many  great  men,  ancient  as  well 
as  recent,  and  on  the  other  hand  f  rom  the  obHgation  im- 
posed  on  his  Oxford  professional  chair,  [^l  determined 
to  undertake  this  same  duty  of  demonstrating  the  oft  men- 
tioned  postulate. 

Now  solely  he  assumes  as  if  certain,  what  foHows: 
namely  that  to  any  given  figure  another  similar  of  any 
magnitude  is  possihle. 

And  that  this  indeed  may  be  presumed  of  any  figure 
(although  in  his  affair  he  assumes  solely  a  rectiHneal  tri- 
angle)  is  weH  argued  from  the  circle,  which  of  course  aU 
admit  can  be  described  with  any-sized  radius. 

Further  the  acute  man  observes  most  cautiously  it 
does  not  thwart  this  his  presumption,  that  besides  the 
equaHty  of  corresponding  angles  also  the  proportionaHty 
of  ah  corresponding  sides  is  required,  in  order  that  a 
rectiHneal  figure,  for  example  a  triangle,  may  be  similar 
to  another  rectiHneal  triangle;  though  stiH  the  definition 
of  proportion,  and  forthwith  of  similar  figures  are  to  be 
taken  f  rom  the  Fifth,  and  the  Sixth  Books  of  EucHd :  For 
(says  he  himself )  Euclid  could  have  put  each  in  front  of 
the  First  Book. 

Hereafter,  this  standing  (which  nevertheless  can  be 
denied  by  any  one,  unless  it  is  demonstrated)  the  famous 
man  carries  out  his  intent  with  reaHy  beautiful  and  in- 
genious  eflFort. 

But  I  am  unwihing  to  fail  in  anything  to  the  charge 
undertaken  by  me. 

Therefore  I  assume  two  triangles,  one  ABC,  and  the 
other  DEF   (fig.  24)   mutuaUy  equiangular.     I  do  not 

los 


similia ;  quia  non  indigeo  proportionalitate  laterum  circa 
angulos  aequales,  immo  neque  ulla  ipsorum  laterum  de- 
terminata  mensura.  Solum  igitur  nolo  triangula  invicem 
aequilatera,  quia  tunc  sufficeret  sola  octava  primi,  sine 
ulla  praesumptione.  Itaque  anguli  ad  puncta  A,  B,  C, 
aequales  sint  angulis  ad  puncta  D,  E,  F ;  sitque  latus  DE 
minus  latere  AB ;  assumaturque  in  AB  portio  AG  aequalis 
ipsi  DE,  atque  item  in  AC  portio  AH  aequalis  ipsi  DF. 
Debere  autem  DF  minorem  esse  ipsa  AC  infra  declarabo. 
Tum  (juncta  GH)  constat  (ex  4.  primi)  aequales  fore 
[41]  angulos  ad  puncta  E,  et  F,  ipsis  AGH,  AHG.  Qua- 
propter ;  cum  modo  dicti  anguli  una  cum  aliis  BGH,  CHG, 
aequales  sint  (ex  13.  primi)  quatuor  rectis;  quatuor  iti- 
dem  rectis  aequales  erunt  anguli  ad  puncta  B,  et  C,  una 
cum  eisdem  angulis  BGH,  CHG.  Igitur  quatuor  simul 
anguli  quadrilateri  BGHC  aequales  erunt  quatuor  rectis; 
ac  propterea  (ex  16.  hujus)  stabilietur  hypothesis  anguli 
recti;  et  simul  (ex  13.  hujus)  Pronunciatum  Euclidaeum. 
Porro  supposui  latus  DF,  sive  AH  sumptum  ipsi 
aequale,  minus  fore  latere  AC.  Si  enim  aequale  foret,  et 
sic  punctum  H  caderet  in  punctum  C ;  tunc  angulus  BCA 
aequalis  foret  (ex  hypothesi)  angulo  EFD,  sive  GCA 
(qui  tunc  fieret)  totum  parti;  quod  est  absurdum.  Sin 
vero  majus  foret,  et  sic  juncta  GH  secaret  in  aliquo 
puncto  ipsam  BC;  jam  angulus  ACB  externus  aequalis 
foret  ex  hypothesi  (contra  16.  primi)  angulo  interno,  et 


te« 


say  wholly  similar;  because  I  do  not  need  the  propor- 

tionality  of  the  sides  about  the  equal  angles,  nay  nor  any 

determinate  measure  of  the  sides 

themselves.     Merely  therefore  I 

do  not  wish  triangles  mutually 

equilateral,  since  then  Eu.  I.  8 

would  alone  suffice,  without  any 

assumption. 

So  let  the  angles  at  the  points  p.     24 

A,  B,  C,  be  equal  to  the  angles 

at  the  points  D,  E,  F;  and  let  the  side  DE  be  less  than 
the  side  AB ;  and  in  AB  is  assumed  the  portion  AG  equal 
to  this  DE,  and  likewise  in  AC  the  portion  AH  equal  to 
this  DF.  But  that  DF  must  be  less  than  AC  I  will  make 
clear  below.  Then  (GH  joined)  follows  (from  Eu.  I.  4) 
the  angles  at  the  points  E,  and  F  will  be  equal  [41]  to  AGH, 
AHG.  However  since  the  just  mentioned  angles,  together 
with  the  others  BGH,  CHG,  are  equal  (Eu.  I.  13)  to  four 
right  angles;  likewise  will  be  equal  to  four  right  angles 
the  angles  at  the  points  B,  and  C,  together  with  these  same 
angles  BGH,  CHG.  Therefore  the  four  angles  of  the 
quadrilateral  BGHC  will  be  together  equal  to  four  right 
angles;  and  consequently  (from  P.  XVI.)  is  estabHshed 
the  hypothesis  of  right  angle;  and  at  the  same  time  (from 
P.  Xni.)  the  Euclidean  postulate. 

Moreover  I  have  supposed  the  side  DF,  or  AH  assumed 
equal  to  it,  to  be  less  than  the  side  AC.  For  if  it  were 
equal,  and  so  the  point  H  should  fall  upon  the  point  C, 
then  the  angle  BCA  would  be  equal  (by  hypothesis)  to 
the  angle  EFD,  or  GCA  (which  then  it  would  become) 
a  part  to  the  whole;  which  is  absurd. 

But  if  it  were  greater,  and  so  the  join  GH  should  cut 
BC  itself  in  some  point,  now  the  external  angle  ACB 
would  be  from  the  hypothesis  equal  (against  Eu.  I.  16)  to 


t07 


opposito  (qui  tunc  fieret)  AHG,  sive  GHA.  Itaque  bene 
supposui  latus  DF  unius  trianguli  minus  fore  latere  AC 
alterius  trianguli,  juxta  hypothesim  jam  stabilitam. 

Quare  ex  duobus  quibusvis  invicem  aequiangulis  tri- 
anguHs,  sed  non  etiam  invicem  aequilateris,  stabilitur 
Pronunciatum  EucHdaeum.     Quod  intendebatur. 

SCHOLION  IV. 

In  quo  exponitur  figuralis  quaedam  exhibitio,  ad  quam 

fortasse  respexit  Euclides,   ut  suum  illud 

Pronunciatum  tanquam  per  se 

notum  stabiliret. 

Praemitto  primo:  sub  quoHbet  angulo  acuto  BAX 
(recole  ex  hac  Tab.  Fig.  12.)  educi  posse  ex  aHquo  [42] 
puncto  X  ipsius  AX  quandam  XB,  quae  sub  quovis  de- 
signato  etiamsi  obtuso  angulo  R,  qui  nimirum  cum  eo 
acuto  BAX  deficiat  a  duobus  rectis;  quandam,  inquam, 
educi  posse  XB,  quae  ad  finitam  distantiam  occurrat  ipsi 
AB  in  quodam  puncto  B.  Nam  id  ipsum  jam  demon- 
stravi  in  SchoHo  post  XHI.  hujus. 

Praemitto  secundo:  eas  AB,  AX  (fig.  25.)  inteUigi 
posse  in  infinitum  protractas  usque  in  quaedam  puncta 
Y,  et  Z;  atque  item  praedictam  XB  (in  infinitum  et  ipsam 
protractam  usque  in  quoddam  punctum  Y)  inteHigi  posse 


ro8 


the  internal  and  opposite  angle  (which  then  would  be- 
come)  AHG,  or  GHA. 

Therefore  I  have  rightly  supposed  the  side  DF  of 
one  triangle  to  be  less  than  the  side  AC  of  the  other 
triangle,  in  accordance  with  the  hypothesis  now  es- 
tabHshed. 

Wherefore  from  any  two  triangles  mutually  equian- 
gular,  but  not  also  mutually  equilateral,  the  EucHdean 
postulate  is  estabHshed.    Quod  intendebatur. 

SCHOLION  IV. 

In  which  is  expounded  on  a  figure  a  certain  consideration 

on  which  Euclid  prohably  thought,  in  order  to  estab- 

lish  that  postulate  of  his  as  per  se  evident. 

I  premise  first :  within  any  acute  angle  BAX  (fig.  12) 
can  be  drawn  from  any  [42]  point  X  of  AX  a  certain 
straight  XB,  under  any  designated  (even  obtuse)  angle 
R  (provided  only  that  R  with  the  acute  BAX  fahs  short 
of  two  right  angles) ;  I  say,  a  certain  XB  can  be  drawn, 
which  at  a  finite  remove  meets  AB  in  a  certain  point  B. 

Y 


For  just  that  I  have  already  demonstrated  in  a  scho- 
Hon  after  P.  XHI. 

I  premise  secondly:  these  AB,  AX  (fig.  25)  can  be 
understood  as  produced  in  infinitum  even  to  certain  points 
Y,  and  Z;  and  Hkewise  the  aforesaid  XB  (produced  in 
infinitum  even  to  a  point  Y)  can  be  understood  to  be 

109 


ita  moveri  super  ea  AZ  versus  partes  puncti  Z,  ut  angulus 
ad  punctum  X  versus  partes  puncti  A  aequalis  semper 
sit  dato  cuivis  obtuso  angulo  R. 

Praemitto  tertio :  nulli  jam  dubitationi  obnoxium  fore 
illud  Pronunciatum  Euclidaeum,  si  antedicta  XY  in  eo 
quantocunque  motu  super  recta  AZ  secet  semper  illam 
AY  in  quibusdam  punctis  B,  H,  D,  P,  atque  ita  conse- 
quenter  in  aliis  punctis  remotioribus  ab  eo  puncto  A.  Ra- 
tio  evidens  est;  quia  sic  duae  quaelibet  in  eodem  plano 
existentes  rectae  AB,  XH,  in  quas  recta  quaelibet  incidens 
AX  duos  ad  easdem  partes  angulos  BAX,  HXA,  duobus 
rectis  minores  efficiat,  convenire  tandem  ad  eas  partes 
deberent  in  unoeodemque  puncto  H. 

Praemitto  quarto:  nulli  item  dubitationi  locum  fore 
super  veritate  praecedentis  hypothetici  assumpti ;  si  poste- 
riores  ilH  externi  anguli  YHD,  YDP,  et  sic  alii  quiHbet 
consequentes,  aut  aequales  semper  sint  priori  externo  an- 
gulo  YBD,  aut  saltem  non  ita  minores  semper  sint,  quin 
eorum  unusquisque  major  semper  sit  parvulo  quopiam  de- 
signato  acuto  angulo  K:  Hoc  enim  stante  manifestum 
fiet,  quod  ea  XY,  in  suo  iHo  quantocunque  motu  versus 
partes  puncti  Z,  nunquam  cessabit  secare  praedictam  AY ; 
quod  utique  (ex  praecedente  notato)  satis  est  ad  sta-[43] 
biHendum  Pronunciatum  controversum. 

Unice  igitur  superest,  ut  quidam  Adversarius  dicat 
angulos  iHos  externos  in  majore,  ac  majore  distantia  ab 
iHo  puncto  A  fieri  semper  minores  sine  ullo  determinato 
Hmite.  Inde  autem  fiet,  ut  iUa  XY  in  suo  iHo  motu  super 
recta  AZ  occurrere  tandem  debeat  ipsi  AY  in  quodam 
puncto  P  sine  uHo  angulo  cum  segmento  PY,  adeo  ut 
nempe  segmentum  ejusmodi  commune  sit  duarum  recta- 


so  moved  above  this  AZ  toward  the  parts  of  the  point  Z, 
that  the  angle  at  the  point  X  toward  the  parts  of  the 
point  A  is  always  equal  to  the  certain  given  obtuse 
angle  R. 

I  premise  thirdly :  that  EucHdean  postulate  would  be 
Hable  now  to  no  doubt,  if  the  aforesaid  XY  in  this  how- 
ever  great  motion  above  the  straight  AZ  cuts  always 
that  AY  in  certain  points  B,  H,  D,  P,  and  so  successively 
in  other  points  more  remote  from  this  point  A. 

The  reason  is  evident;  since  thus  any  two  straights 
AB,  XH  lying  in  the  same  plane,  upon  which  any  straight 
AX  cutting  makes  two  angles  toward  the  same  parts 
BAX,  HXA,  less  than  two  right  angles,  must  at  length 
meet  toward  those  parts  in  one  and  the  same  point  H. 

I  premise  fourthly :  hkewise  will  be  no  doubt  about 
the  truth  of  the  preceding  hypothetical  assumption,  if 
the  later  external  angles  YHD,  YDP  and  so  any  other 
succeeding  ones,  either  always  are  equal  to  the  preceding 
external  angle  YBD,  or  at  least  always  will  be  not  so 
much  less  but  that  any  one  of  them  always  will  be  greater 
than  any  little  designated  acute  angle  K.  For,  this  hold- 
ing,  it  is  manifest  that  this  XY  in  that  however  great  mo- 
tion  of  its  toward  the  parts  of  the  point  Z,  never  will  cease 
to  cut  the  af oresaid  AY ;  which  assuredly  ( f rom  the  pre- 
ceding  remark)  is  sufficient  for  establishing  [43]  the  con- 
troverted  postulate. 

Solely  therefore  remains,  that  some  adversary  may 
say  those  external  angles  at  greater  and  greater  distance 
from  the  point  A  may  become  always  less  without  any 
determinate  limit. 

But  thence  would  follow,  that  XY  in  its  motion  above 
the  straight  AZ  would  at  length  meet  AY  in  a  certain 
point  P  without  any  angle  with  the  segment  PY,  so  that 
indeed  a  segment  of  the  two  straights  APY,  and  XPY 


"f 


rum  APY,  et  XPY.    At  hoc  evidenter  repugnat  naturae 
lineae  rectae. 

Sin  vero  cuiquam  minus  opportunus  videatur  angu- 
lus  obtusus  ad  illud  punctum  X  versus  partes  puncti  A, 
nullo  negotio  supponi  poterit  rectus;  adeo  ut  nempe  (in 
motu  praedictae  XY  ad  angulos  semper  rectos  super  recta 
AZ)  manifestius  appareat  singula  illius  X Y  puncta  aequa- 
biliter  semper  moveri  relate  ad  subjectam  AZ;  ac  prop- 
terea  nequire  jam  dictam  XY  transire  de  secante  in  non 
secantem  alterius  indefinitae  AY,  nisi  eam  aut  aliquando 
in  aliquo  puncto  praecise  contingat,  aut  ipsi  occurrat  in  ali- 
quo  puncto  P,  ubi  cum  eadem  AY  commune  obtineat 
segmentum  PY;  quorum  utrunque  adversari  naturae 
lineae  rectae  ostendam  ad  XXXIII.  hujus.  Igitur  juxta 
veram  ideam  Hneae  rectae,  debebit  illa  XY,  in  quanta- 
cunque  distantia  puncti  X  a  puncto  A,  occurrere  semper 
in  aHquo  puncto  ipsi  AY.  Atque  id  quidem  (quantumhbet 
parvus  supponatur  acutus  angulus  ad  punctum  A)  satis 
esse  ad  demonstrandum,  contra  hypothesim  anguli  acuti, 
Pronunciatum  Euclidaeum,  constabit  ex  XXVII.  hujus. 

PROPOSITIO  XXII. 

Si  duae  rectae  AB,  CD  in  eodem  plano  existentes  perpen- 
diculariter  insistant  cuidam  rectae  BD;  ipsa  autem 
AC  jungens  ea  perpendicida  internos  (in  hypothesi 
angidi  acuti)  acu-l^^]tos  angidos  cum  eisdem  efficiat: 
Dico  (fig.  26.)  rectas  terminatas  AC,  BD  commune 
aliquod  habere  perpendicidum,  et  quidem  intra  limi- 
tes  designatis  punctis  A,  et  C  praefinitos. 


would  be  in  this  way  common.  But  this  is  evidently 
repugnant  to  the  nature  of  the  straight  line. 

But  if  to  any  one  may  seem  less  opportune  the  obtuse 
angle  at  the  point  X  toward  the  parts  of  the  point  A,  it 
may  easily  be  supposed  right;  so  that  indeed  (in  the 
motion  of  the  aforesaid  XY  at  angles  always  right  above 
the  straight  AZ)  more  manifestly  may  appear  that  the 
single  points  of  that  XY  are  always  moved  uniformly 
relatively  to  the  basal  AZ;  and  therefore  the  aforesaid 
XY  cannot  go  over  from  a  secant  into  a  non-secant  of  the 
other  indefinite  AY,  unless  either  once  in  some  point  it 
precisely  touches  it,  or  meets  it  in  some  point  P,  where 
it  has  with  this  AY  a  common  segment  PY;  each  of 
which  I  shall  show  contrary  to  the  nature  of  the  straight 
Hne  in  P.  XXXIII. 

Therefore  in  accordance  with  the  true  idea  of  the 
straight  Hne,  must  that  XY,  however  great  the  distance 
of  the  point  X  from  the  point  A,  always  meet  in  some 
point  this  AY.  And  that  this  indeed  (however  small  is 
supposed  the  acute  angle  at  the  point  A)  is  sufficient  for 
demonstrating,  against  the  hypothesis  of  acute  angle,  the 
EucHdean  postulate,  will  follow  from  P.  XXVII. 

PROPOSITION  XXII. 

//  two  straights  AB,  CD  existing  in  the  same  plane  stand 
perpendicular  to  a  certain  straight  BD ;  hut  AC  join- 

5 


ing  these  perpendiculars  makes  a 

with  them  internal  acute  angles  h 

{in  hypothesis  of  acute  angle)  :  ^ 

[44]  /  say  (fig.  26)  the  termi-  x- 

nated  straights  AC,  BD  have  a  <»i 
common  perpendicular,  and  in-  Fig.  26. 

deed  within  the  limits  fixed  by  the  designated  points 
A  and  C. 


-lO 


tX3 


Demonstratur.  Si  enim  aequales  sint  ipsae  AB,  CD ; 
constat  (ex  2.  hujus)  rectam  LK,  a  qua  bifariam  dividan- 
tur  illae  duae  AC,  et  BD,  commune  fore  eisdem  perpen- 
diculum.  Sin  vero  alterutra  sit  major,  ut  puta  AB :  de- 
mittatur  ad  BD  (juxta  12.  primi)  ex  quovis  puncto  L 
ipsius  AC  perpendicularis  LK,  occurrens  alteri  BD  in  K. 
Occurret  autem  in  aliquo  puncto  K,  consistente  inter 
puncta  B,  et  D;  ne  (contra  17.  primi)  perpendicularis 
LK  secet  alterutram  AB,  aut  CD,  perpendiculares  eidem 
BD.  Si  ergo  anguli  ad  punctum  L  recti  non  sunt,  unus 
eorum  acutus  erit,  et  alter  obtusus.  Sit  obtusus  versus 
punctum  C.  Jam  vero  intelligatur  LK  ita  procedere  ver- 
sus  AB,  ut  semper  ad  rectos  angulos  insistat  ipsi  BD,  at- 
que  item  opportune  aucta,  aut  imminuta,  in  aliquo  sui 
puncto  secet  rectam  AC.  Constat  angulos  ad  puncta  inter- 
sectiva  ipsius  AC  non  posse  omnes  esse  obtusos  versus 
partes  puncti  C,  ne  tandem  in  ipso  puncto  A,  dum  recta 
LK  congruet  cum  recta  AB,  angulus  ad  punctum  A  ver- 
sus  partes  puncti  C  sit  obtusus,  cum  ad  eas  partes  positus 
sit  acutus.  Quoniam  ergo  angulus  ad  punctum  L  ipsius 
LK  positus  est  obtusus  versus  partes  puncti  C,  non  trans- 
ibit  in  eo  motu  recta  LK  ad  faciendum  in  aliquo  sui 
puncto  cum  recta  AC  angulum  acutum  versus  partes  prae- 
dicti  puncti  C,  nisi  prius  transeat  ad  constituendum  in 
aliquo  sui  puncto  cum  eadem  AC  angulum  rectum  versus 
partes  ejusdem  puncti  C.  Erit  igitur  inter  puncta  A,  et 
L  unum  aliquod  punctum  intermedium  H,  in  quo  HK 
perpendicularis  ipsi  BD  sit  etiam  perpendicularis  alteri 
AC.  ■••  :\!'^ 

Simili  modo  ostendetur  adesse  aliquam  XK  inter  ipsas 
LK,  CD,  quae  sit  perpendicularis  et  rectae  BD,  et  [45] 


"4 


Proof.  For  if  AB,  CD  are  equal,  it  follows  (from 
P.  II.)  that  the  straight  LK,  by  which  these  two  AC  and 
BD  are  bisected,  will  be  to  them  a  common  perpendicular. 
But  if  either  be  the  greater,  as  suppose  AB;  let  fall  to 
BD  (according  to  Eu.  I.  12)  from  any  point  L  of  AC 
the  perpendicular  LK,  meeting  the  other  BD  in  K. 

But  it  will  meet  it  in  some  point  K  existing  between 
the  points  B  and  D;  otherwise  (contrary  to  Eu.  I.  17) 
the  perpendicular  LK  would  cut  either  AB,  or  CD,  per- 
pendicular  to  the  same  BD.  If  then  the  angles  at  the 
point  L  are  not  right,  one  of  them  will  be  acute  and  the 
other  obtuse. 

Let  the  obtuse  be  toward  the  point  C.  But  now  LK 
is  understood  so  to  proceed  toward  AB,  that  it  always 
stands  at  right  angles  to  BD,  and  likewise  opportunely 
increased,  or  diminished,  in  some  point  of  it  cuts  the 
straight  AC.  It  f ollows  that  the  angles  at  the  intersection 
points  with  AC  cannot  all  be  obtuse  toward  the  parts  of 
the  point  C,  lest  at  length  in  that  point  A,  where  the 
straight  LK  is  congruent  with  the  straight  AB,  the  angle 
at.  the  point  A  toward  the  parts  of  the  point  C  should  be 
obtuse,  when  toward  these  parts  it  is  by  hypothesis  acute. 

Since  therefore  the  angle  at  the  point  L  of  this  LK 
is  by  hypothesis  obtuse  toward  the  parts  of  the  point  C, 
the  straight  LK  will  not  change  over  in  this  motion  so  as 
to  make  in  some  point  of  it  with  the  straight  AC  an  angle 
acute  toward  the  parts  of  the  aforesaid  point  C,  unless 
previously  it  changes  over  so  as  to  make  in  some  point  of  it 
with  this  AC  an  angle  right  toward  the  parts  of  this  same 
point  C. 

Therefore  between  the  points  A,  and  L  will  be  some 
one  intermediate  point  H,  in  which  HK  perpendicular  to 
this  BD  is  also  perpendicular  to  the  other  AC. 

In  a  similar  manner  is  shown  to  be  present  a  certain 
XK  between  LK,  CD,  which  is  perpendicular  both  to  the 

"5 


rectae  AC,  dum  scilicet  angulus  obtusus  ad  punctum  L  po- 
natur  consistere  versus  partes  puncti  A. 

Constat  igitur  rectas  AC,  BD  commune  aliquod  ha- 
bituras  esse  perpendiculum,  et  quidem  intra  limites  desig- 
natis  punctis  A,  et  C  praefinitos,  quoties  junctae  AB,  CD 
in  eodem  plano  existant,  sintque  perpendiculares  ipsi  BD. 
Quod  erat  etc. 

PROPOSITIO  XXIII. 

Si  duae  quaelihet  rectae  AX,  BX  (fig.  27.)  in  eodem 
plano  existant;  vel  unum  aliquod  (etiam  in  hypothesi 
anguli  acuti)  commune  obtinent  perpendiculum;  vel 
in  alterutram  eandem  partem  protractae,  nisi  all- 
quando  ad  finitam  distantiam  una  in  alteram  incidat, 
semper  magis  ad  se  invicem  accedunt. 

Demonstratur.  Ex  quolibet  puncto  A  ipsius  AX  de- 
mittatur  ad  rectam  BX  perpendicularis  AB.  Si  ipsa  BA 
efficiat  cum  AX  angulum  rectum,  habemus  intentum 
communis  perpendiculi.  Caeterum  vero  ea  recta  efficiat 
ad  alterutram  partem,  ut  puta  versus  partes  puncti  X,  an- 
gulum  acutum.  Itaque  in  praedicta  recta  AX  designen- 
tur  inter  puncta  A,  et  X  quaelibet  puncta  D,  H,  L,  ex 
quibus  demittantur  ad  rectam  BX  perpendiculares  DK, 
HK,  LK.  Si  unus  aliquis  angulus  ad  puncta  D,  H,  L  acu- 
tus  sit  versus  partes  puncti  A,  constat  (ex  praecedente) 
unum  aliquod  adfuturum  commune  perpendiculum  ipsa- 
rum  AX,  BX.  Sin  vero  omnis  hujusmodi  angulus  sit 
major  acuto;  vel  unus  aliquis  erit  rectus,  et  sic  rursum 


Zli 


straight  BD,  and  [45]  to  the  straight  AC,  if  namely  an 
angle  at  the  point  L  is  assumed  to  be  obtuse  toward  the 
parts  of  the  point  A. 

It  follows  therefore  that  the  straights  AC,  BD,  will 
have  a  common  perpendicular,  and  indeed  within  the 
limits  fixed  by  the  designated  points  A,  and  C,  when  the 
joins  AB,  CD  exist  in  the  same  plane  and  are  perpen- 
dicular  to  BD. 

Quod  erat  etc. 

PROPOSITION  XXIII. 

//  any  two  straights  AX,  BX  (fig.  27)  are  in  the  sanie 
plane;  either  they  have  (even  in  the  hypothesis  of 
acute  angle)  a  common  perpendicular ;  or  prolonged 
toward  either  the  same  part,  unless  somewhere  at  a 
finite  distance  one  meets  the  other,  they  mutually 
approach  ever  more  toward  each  other. 

Proof.  From  any  point  A  of  AX  let  fall  to  the 
straight  BX  the  perpendicular  AB.  If  BA  makes  with 
AX  a  right  angle,  we  have  the  as- 
serted  case  of  a  common  perpendicu- 
lar.  But  otherwise  this  straight  makes 
toward  one  or  the  other  part,  as  sup- 
pose  toward  the  parts  of  the  point  X, 
an  acute  angle.  Accordingly  in  the 
aforesaid  straight  AX  between  the 
points  A  and  X  any  points  D,  H,  L 
are  designated,  f  rom  which  are  let  f  all  ^*^*  ^^ 

to  the  straight  BX  the  perpendiculars  DK,  HK,  LK. 

If  any  one  angle  at  the  points  D,  H,  L  be  acute  toward 
the  parts  of  the  point  A,  it  follows  (from  the  preceding) 
that  AX,  BX  will  have  a  common  perpendicular. 

But  if  every  angle  of  this  sort  be  greater  than  acute ; 
either  some  one  will  be  right,  and  thus  again  we  shall 

»»7 


habemus  intentum  communis  perpendiculi,  cum  omnes 
anguli  ad  puncta  K  supponantur  recti ;  vel  omnes  illi  an- 
guli  ponuntur  obtusi  versus  partes  puncti  A,  ac  propterea 
omnes  itidem  acuti  versus  partes  puncti  X,  et  sic  rursum 
argumentor.  Quoniam  in  quadrilatero  KDHK  recti  sunt 
[46]  anguli  ad  puncta  K,  ponitur  autem  acutus  angulus  ad 
punctum  D,  erit  (ex  Cor.  11.  post  3.  hujus)  latus  DK 
majus  latere  HK.  SimiH  modo  ostendetur  latus  HK 
majus  esse  latere  LK ;  atque  ita  semper,  conferendo  inter 
se  perpendiculares  ex  quoHbet  puncto  altiore  ipsius  AX 
demissas  ad  alteram  BX.  Quapropter  ipsae  AX,  BX 
semper  magis  versus  partes  puncti  X  ad  se  invicem  acce- 
dent :  Quae  est  altera  pars  propositi  disjuncti. 

Ex  quibus  omnibus  constat  duas  quasHbet  rectas  AX, 
BX,  quae  in  eodem  plano  existant,  vel  unum  aHquod 
(etiam  in  hypothesi  anguH  acuti)  commune  habere  per- 
pendiculum,  vel  in  alterutram  eandem  partem  protractas, 
nisi  aHquando  ad  iinitam  distantiam  una  in  alteram  inci- 
dat,  semper  magis  ad  se  invicem  accedere.   Quod  erat  etc. 

COROLLARIUM  L 

Hinc  anguH  versus  basim  AB  erunt  semper  obtusi  ad 
illud  punctum  ipsius  AX,  ex  quo  demittitur  perpendicu- 
laris  ad  rectam  BX :  erunt,  inquam,  semper  obtusi,  quoties 
duae  iHae  AX,  et  BX  semper  magis  ad  se  invicem  acce- 
dant  versus  partes  punctorum  X ;  quod  quidem  sano  modo 
inteHigi  debet,  nimirum  de  perpendicularibus  demissis 
ante  praedictum  occursum,  si  forte  ad  finitam  distantiam 
una  in  alteram  incidere  debeat. 


ii8 


have  the  asserted  case  of  a  common  perpendicular,  since 
all  angles  at  the  points  K  are  supposed  right ;  or  all  those 
angles  toward  the  parts  of  the  point  A  are  obtuse,  and 
therefore  all  therewith  acute  toward  the  parts  of  the 
point  X,  and  so  again  I  argue :  Since  in  the  quadrilateral 
KDHK  the  angles  at  the  points  K  are  right,  [46]  but  the 
angle  at  the  point  D  is  acute,  the  side  DK  will  be  (from 
Cor.  II.  to  P.  III.)  greater  than  the  side  HK. 

In  a  similar  way  the  side  HK  is  shown  to  be  greater 
than  the  side  LK;  and  so  always,  comparing  to  each 
other  perpendiculars  from  any  ever  higher  points  of  AX 
let  fall  upon  the  other  BX. 

Wherefore  AX,  BX  mutually  approach  each  other 
ever  more  toward  the  parts  of  the  point  X :  which  is  the 
second  part  of  the  disjunct  proposition. 

From  all  which  follows  that  any  two  straights  AX, 
BX,  which  are  in  the  same  plane,  either  have  (even  in 
the  hypothesis  of  acute  angle)  a  common  perpendicular, 
or  produced  toward  either  the  same  part,  unless  some- 
where  at  a  finite  distance  one  meets  the  other,  mutually 
approach  each  other  ever  more. 

Quod  erat  etc. 

COROLLARY  I. 

Hence  the  angles  toward  the  base  AB  will  be  always 
obtuse  at  each  point  of  AX,  from  which  is  let  fall  a 
perpendicular  to  the  straight  BX:  will  be,  I  say,  always 
obtuse,  as  those  two  AX,  and  BX  mutually  approach 
each  other  ever  more  toward  the  parts  of  the  points  X; 
which  of  course  should  be  understood  in  a  sane  way,  of 
perpendiculars  let  fall  before  the  mentioned  meeting,  if 
perchance  one  is  to  strike  upon  the  other  at  a  finite  dis- 
tance. 


»9 


SCHOLION. 

Video  tamen  inquiri  hic  posse,  qua  ratione  ostenden- 
dum  sit  commune  illud  perpendiculum ;  quoties  recta  quae- 
piam  PFHD  (fig.  28.)  occurrens  duabus  AX,  BX  in 
punctis  F,  et  H,  duos  ad  easdem  partes  efficiat  intemos 
angulos  AHF,  BFH,  non  eos  quidem  rectos,  sed  tamen 
[47]  aequales  simul  duobus  rectis.  Ecce  autem  commune 
illud  perpendiculum  geometrice  demonstratum.  Divisa 
FH  bifariam  in  M  demittantur  ad  AX,  et  BX  perpendicu- 
lares  MK,  ML.  Angulus  MFL  aequalis  erit  (ex  13, 
primi)  angulo  MHK,  qui  nempe  supponitur  duos  rectos 
efficere  cum  angulo  BFH.  Praeterea  recti  sunt  anguli  ad 
puncta  K,  et  L ;  ac  rursum  aequales  sunt  ipsae  MF,  MH. 
Igitur  (ex  26.  primi)  aequales  itidem  erunt  anguli  FML, 
HMK.  Quare  angulus  HMK  duos  efficiet  rectos  angulos 
cum  angulo  HML,  prout  cum  eodem  duos  efficit  rectos 
angulos  (ex  13.  primi)  angulus  FML.  Igitur  (ex  14. 
primi)  una  erit  recta  linea  continuata  ipsa  KML,  com- 
mune  idcirco  perpendiculum  praedictis  rectis  AX,  BX. 
Quod  erat  etc. 

»     COROLLARIUM  11. 

Sed  rursum  docere  hinc  possum,  quod  illae  duae  AX, 
BX,  in  quas  incidens  recta  PFHD,  aut  duos  efficiat  cum 
ipsis  AX,  BX  internos  ad  easdem  partes  angulos  aequales 


SCHOLION. 

I  see  indeed  it  may  here  be  asked  in  what  way  that 
common  perpendicular  can  be  shown,  when  any  straight 
PFHD  (fig.  28)  meeting  two  AX,  BX  in  points  F,  and 
H,  makes  toward  the  same  parts  two  internal  angles 
AHF,  BFH,  not  themselves  indeed  right,  but  neverthe- 
less  [47]  together  equal  to  two  rights.  But  behold  that 
common  perpendicular  geometrically  demonstrated. 


Fig.  28. 

FH  being  bisected  in  M,  perpendiculars  MK,  ML 
are  let  fall  to  AX  and  BX.  The  angle  MFL  will  be  equal 
(Eu.  I.  13)  to  the  angle  MHK,  which  indeed  is  assumed 
to  make  up  two  right  angles  with  the  angle  BFH.  More- 
over  the  angles  at  the  points  K,  and  L  are  right ;  and  again 
MF,  MH  are  equal.  Therefore  (Eu.  L  26)  so  are  the 
angles  FML,  HMK  equal.  Wherefore  the  angle  HMK 
makes  two  right  angles  with  the  angle  HML,  since  with 
this  the  angle  FML  (Eu.  I.  13)  makes  two  right  angles. 
Therefore  (Eu.  L  14)  KML  will  be  in  one  continuous 
straight  line,  consequently  a  common  perpendicular  to 
the  aforesaid  straights  AX,  BX. 

Quod  erat  etc. 

COROLLARY  IL 

But  again  I  am  able  hence  to  show  that  those  two 
straights  AX,  BX,  meeting  with  which  the  straight  PFHD 
makes  with  the  said  AX,  BX  either  two  internal  angles 
toward  the  same  parts  equal  to  two  right  angles,   or 


duobus  rectis;  aut  consequenter  (ex  13.  et  15.  primi)  al- 
ternos  sive  externos,  sive  internos  angulos  inter  se  aequa- 
les;  aut  rursum,  eodem  titulo,  externum  (ut  puta  DHX) 
aequalem  interno,  et  opposito  HFX :  quod,  inquam,  illae 
duae  rectae  neque  ad  infinitam  earundem  productionem 
coire  inter  se  possint.  Si  enim  ex  quolibet  puncto  N  ipsius 
AX  demittatur  ad  BX  perpendicularis  NR,  erit  haec  in 
ipsa  hypothesi  anguH  acuti  (quae  utique  sola  obesse  nobis 
posset)  major  (ex  Cor.  I.  post  3.  hujus)  eo  communi  per- 
pendiculo  KL.  Non  igitur  illae  duae  AX,  BX,  convenire 
unquam  inter  se  poterunt. 

Porro  autem  demonstratas  hinc  habes  Propos.  27.  et 
28.  Libri  primi  EucHdis ;  et  quidem  citra  immediatam  de- 
pendentiam  a  praecedentibus  16.  et  17.  ejusdem  primi, 
cir-[48]ca  quas  oriri  posset  difficultas,  quoties  sub  basi  finita 
infinitilaterum  esset  triangulum ;  ad  quale  nempe  triangu- 
lum  provocare  non  dubitaret,  qui  eas  duas  AX,  BX  ad 
infinitam  saltem  distantiam  inter  se  coituras  censeret, 
quamvis  anguH  ad  incidentem  PFHD  tales  forent,  quales 
supposuimus. 

Praeterea,  propter  demonstratum  commune  perpen- 
diculum  KL,  nequirent  sane  ihae  duae  KX,  LX  ad  suam 
partem  punctorum  X  simul  concurrere,  quin  etiam  (ex 
faciH  inteHecta  superpositione)  ad  alteram  etiam  partem 
simul  concurrerent  reliquae  et  ipsae  interminatae  KA, 
LB.  Quare  duae  rectae  AX,  BX  clauderent  spatium; 
quod  est  contra  naturam  lineae  rectae. 

Sed  haec  posteriora  sunt.  Nam  in  praecedentibus  nus- 
quam  adhibui  aut  16.  aut  17.  primi,  nisi  ubi  clare  agere- 
tur  de  triangulo  omni  ex  parte  circumscripto,  prout  nempe 
in  Proemio  ad  Lectorem  ita  me  curaturum  spoponderam. 


consequently  (from  Eii.  I.  13  and  15)  alternate  external 
or  internal  angles  equal  to  one  another,  or  again,  f  rom  the 
same  cause,  an  external  (as  suppose  DHX)  equal  to  an  in- 
ternal  and  opposite  HFX ;  that,  say  I,  those  two  straights 
not  even  in  their  infinite  production  can  meet  one  another. 

For  if  from  any  point  N  of  AX  is  let  fall  to  BX  the 
perpendicular  NR,  this  will  be  in  the  hypothesis  of  acute 
angle  (which  alone  in  any  case  can  hinder  us)  greater 
(from  P.  ni.,  Cor.  I.)  than  the  common  perpendicular 
KL.  Therefore  those  two  straights  AX,  BX  cannot  ever 
meet  one  another. 

But  furthermore  here  you  have  Eu.  I.  27  and  28 
demonstrated,  and  indeed  without  immediate  dependence 
from  the  preceding  16  and  17  of  the  same  First  Book, 
about  [48]  which  difficulties  could  arise  when  the  triangle 
should  be  of  infinite  sides  on  a  finite  base;  to  which  sort 
of  a  triangle  without  doubt  would  refer  one  who  believed 
that  these  two  straights  AX,  BX  met  one  another  at  least 
at  an  infinite  distance,  although  the  angles  at  the  trans- 
versal  PFHD  were  such  as  we  have  supposed. 

Moreover,  on  account  of  the  demonstrated  common 
perpendicular  KL,  surely  those  two  KX,  LX  cannot  come 
together  toward  the  part  of  the  points  X,  since  also  (from 
a  superposition  easily  understood)  toward  the  other  part 
also  would  meet  at  the  same  time  the  remaining  and 
themselves  unterminated  KA,  LB.  Wherefore  two 
straights  AX,  BX  would  enclose  a  space;  which  is  con- 
trary  to  the  nature  of  the  straight  line. 

But  these  things  are  later.  For  in  the  preceding  I 
have  never  applied  either  Eu.  I.  16  or  17,  except  where 
clearly  it  treats  of  a  triangle  bounded  on  every  side,  as 
indeed  I  promised  I  would  so  take  care  to  do  in  the 
Preface  to  the  Reader. 


123 


PROPOSITIO  XXIV. 

lisdent  manentibus:  Dico  quatuor  simul  angulos  (fig.27.) 
quadrilateri  KDHK  proximioris  basi  AB  minore^ 
esse  (in  hypothesi  anguli  acuti)  quatuor  simul  atv- 
gulis  quadrilateri  KHLK  remotioris  ab  eadem  basi, 
atque  ita  quidem,  sive  illae  duae  AX,  BX  aliquando 
ad  finitam  distantiam  incidant  versus  partes  puncti 
X;  sive  nunquam  inter  se  incidant;  sed  versus  eas 
partes  aut  semper  magis  ad  se  invicem  accedant,  aut 
aliquando  recipiant  commune  perpendiculum,  post 
quod  nempe  (juxta  Cor.  //.  praec.  Propos.)  ad  eas- 
dem  partes  incipiant  invicem  dissilire. 

Demonstratur.  Verum  hic  supponimus  portiones  KK 
sumptas  esse  invicem  aequales.  Quoniam  igitur  (ex  prae- 
[49]cedente)  latus  DK  majus  est  latere  HK,  ac  similiter 
HK  majus  latere  LK ;  sumatur  in  HK  portio  MK  aequalis 
ipsi  LK,  et  in  DK  portio  NK  aequalis  ipsi  HK;  jungan- 
turque  MN,  MK,  LK ;  nimirum  punctum  K  intermedium 
cum  puncto  L,  et  punctum  K  vicinius  puncto  B  cum 
puncto  M.  Jam  sic  progredior.  Quandoquidem  latera 
trianguli  KKL  (initium  semper  ducam  a  puncto  K  vici- 
niore  puncto  B)  aequalia  sunt  lateribus  trianguli  KKM, 
et  anguli  comprehensi  aequales,  utpote  recti;  aequales 
etiam  erunt  (ex  4.  primi)  bases  LK,  MK;  atque  item 
aequales,  qui  correspondent  invicem  anguli,  ad  easdem 


ia4 


PROPOSITION  XXIV. 

The  same  remaining:  I  say  the  four  angles  together 
(fig.  27)  of  the  qmdrilateral  KDHK  nearer  the 
base  AB  are  less  (in  the  hypothesis  of  acute  angle) 
than  the  four  angles  together  of  the  quadrilateral 
KHLK  more  remote  from  the  same  hase\  and  in- 
deed  this  is  sOj  whether  those  two  AX,  BX  some- 
where  at  a  finite  distance  meet  toward  the  parts  of 
the  point  X ;  or  never  meet  one 
another;  but  toward  those  parts 
either  ever  more  mutually  ap- 
proach  each  other,  or  some- 
where  receive  a  common  per- 
pendicular,  after  which  of 
course  (in  accordance  with  Cor. 
//.  of  the  preceding  proposi- 
tion)    toward   the  same  parts  ^^8-  27. 

they  begin  mutually  to  separate. 

Proof.  Here  however  we  suppose  the  portions  KK 
assumed  to  be  mutually  equal.  Since  therefore  (from 
the  preceding)  [49]  the  side  DK  is  greater  than  the  side 
HK,  and  similarly  HK  greater  than  the  side  LK,  the 
portion  MK  in  HK  is  assumed  equal  to  LK,  and  in 
DK  the  portion  NK  equal  to  HK;  and  MN,  MK,  LK 
are  joined,  truly  the  intermediate  point  K  with  the  point 
L,  and  the  point  K  nearer  to  the  point  B  with  the  point  M. 

Now  I  proceed  thus. 

Since  indeed  the  sides  of  the  triangle  KKL  (I  make 
beginning  always  from  the  point  K  nearer  to  the  point  B) 
are  equal  to  the  sides  of  the  triangle  KKM,  and  the  in- 
cluded  angles  equal,  as  being  right,  equal  also  will  be 
(from  Eu.  I.  4)  the  bases  LK,  MK,  and  likewise  equal 
the  angles  which  correspond  mutually,  at  these  bases, 


bases,  nimirum  angulus  KLK  angulo  KMK,  et  angulus 
LKK  angulo  MKK.  Igitur  aequales  etiam  sunt  residui 
NKM,  et  HKL.  Quare,  cum  latera  NK,  KM,  trianguli 
NKM  aequalia  itidem  sint  lateribus  HK,  KL  trianguli 
HKL;  aequales  etiam  erunt  (ex  eadem  4.  primi)  bases 
NM,  HL;  anguli  KNM,  KHL;  ac  tandem  anguli  KMN, 
KLH.  Sunt  autem  in  prioribus  triangulis  jam  probati 
aequales  anguli  KLK,  et  KMK.  Igitur  totus  angulus 
NMK  aequalis  est  toti  angulo  HLK.  Quare,  cum  omnes 
ad  puncta  K  anguli  sint  recti,  manifeste  consequitur  om- 
nes  simul  quatuor  angulos  quadrilateri  KNMK  aequales 
esse  omnibus  simul  quatuor  angulis  quadrilateri  KHLK. 
Quoniam  vero  duo  simul  anguli  ad  puncta  N,  et  M  in 
quadrilatero  KNMK  majores  sunt,  in  hypothesi  anguli 
acuti,  duobus  simul  angulis  (ex  Cor.  post  XVI.  hujus)  ad 
puncta  D,  et  H  in  quadrilatero  NDHM,  seu  quadrilatero 
KDHK;  consequens  inde  est,  ut  (additis  communibus 
rectis  angulis  ad  puncta  K)  quatuor  simul  anguli  quadri- 
lateri  KNMK,  seu  quadrilateri  KHLK,  majores  sint  (in 
hypothesi  anguli  acuti)  quatuor  simul  angulis  quadrilateri 
KDHK.     Quod  erat  demonstrandum.  [50] 

COROLLARIUM. 

Sed  opportune  observari  hic  debet,  nihil  defuturum 
factae  argumentationi,  quamvis  angulus  ad  punctum  L 
poneretur  rectus,  juxta  hypothesin  anguli  acuti.  Nam 
adhuc  illa  communis  perpendicularis  LK  minor  foret  (ex 
Cor.  I.  post  III.  hujus)  altera  perpendiculari  HK,  ex  qua 
propterea  sumi  adhuc  posset  portio  MK  aequalis  prae- 


m6 


indeed  the  angle  KLK  to  the  angle  KMK,  and  the  angle 
LKK  to  the  angle  MKK.  Therefore  equal  also  are  the 
remainders  NKM  and  HKL.  Wherefore,  since  the  sides 
NK,  KM  of  the  triangle  NKM  are  equal  in  the  same  way 
to  the  sides  HK,  KL  of  the  triangle  HKL,  equal  also 
will  be  (from  the  same  Eu.  I.  4)  the  bases  NM,  HL, 
the  angles  KNM,  KHL,  and  finally  the  angles  KMN, 
KLH.  But  in  the  preceding  triangles  are  already  proved 
equal  the  angles  KLK,  KMK.  Therefore  the  whole  angle 
NMK  is  equal  to  the  whole  angle  HLK. 

Wherefore,  since  all  angles  at  the  points  K  are  right, 
it  follows  manifestly  all  four  angles  together  of  the 
quadrilateral  KNMK  are  equal  to  all  four  angles  together 
of  the  quadrilateral  KHLK. 

But  since  the  two  angles  together  at  the  points  N  and 
M  in  the  quadrilateral  KNMK  are  greater,  in  hypothesis 
of  acute  angle,  than  the  two  angles  together  (from  Cor. 
after  P.  XVL)  at  the  points  D  and  H  in  the  quadrilateral 
NDHM,  or  the  quadrilateral  KDHK,  the  consequence 
thence  is,  that  (the  common  right  angles  at  the  points  K 
being  added)  the  four  angles  together  of  the  quadrilateral 
KNMK,  or  the  quadrilateral  KHLK  are  greater  (in 
hypothesis  of  acute  angle)  than  the  four  angles  together 
of  the  quadrilateral  KDHK. 

Quod  erat  demonstrandum.  [50] 

COROLLARY. 

But  it  ought  here  opportunely  to  be  observed,  nothing 
will  fail  in  the  argument  made,  although  the  angle  at  the 
point  L  is  assumed  right,  together  with  hypothesis  of 
acute  angle.  For  still  that  common  perpendicular  LK 
would  be  less  (from  Cor.  I.  to  P.  IH.)  than  the  other 
perpendicular  HK,  from  which  therefore  still  a  portion 
MK  could  be  assumed  equal  to  the  aforesaid  LK. 


dictae  LK :  Quo  stante  constat  nullum  posse  obicem  inter- 
currere. 

SCHOLION. 

Dubitari  nihilominus  posset,  an  ex  quolibet  puncto  K 
(assumpto  nimirum  in  BX  ante  occursum  ipsius  BX  in 
alteram  AX)  perpendicularis  educta  versus  partes  rectae 
AX  occurrere  huic  debeat  (fig.  29.)  in  aliquo  puncto  L; 
dum  nempe  illae  duae,  ante  praedictum  occursum,  ponan- 
tur  ad  se  invicem  semper  magis  accedere.  Ego  autem 
dico  ita  omnino  secuturum. 

Demonstratur.  Assignatum  sit  in  BX  quodvis  punc- 
tum  K.  Sumatur  in  AX  quaedam  AM  aequalis  summae 
ex  ipsa  BK,  et  dupla  AB.  Tum  ex  puncto  M  ducatur  ad 
BX  (juxta  12.  primi)  perpendicularis  MN.  Erit  MN 
(juxta  praesentem  suppositionem)  minor  ipsa  AB.  Quare 
AM  (facta  aequalis  summae  ex  ipsa  BK,  et  dupla  AB) 
major  erit  sumnia  ipsarum  BK,  AB,  et  NM.  Jam  osten- 
dere  oportet  eandem  AM  minorem  esse  summa  ipsarum 
BN,  AB,  et  MN,  ut  inde  constet  eam  BN  majorem  esse 
praedicta  BK,  ac  propterea  punctum  K  jacere  inter  puncta 
B,  et  N.  Jungatur  BM.  Erit  latus  AM  (ex  20.  primi) 
minus  duobus  simul  reHquis  lateribus  AB,  et  BM.  Rur- 
sum  [51]  latus  BM  (ex  eadem  20.  primi)  minus  erit  duo- 
bus  simul  lateribus  BN,  et  MN.  Igitur  latus  AM  multo 
minus  erit  tribus  simul  lateribus  AB,  BN,  et  NM.  Hoc 
autem  erat  ostendendum,  ut  constaret  punctum  K  jacere 
inter  puncta  B,  et  N.    Inde  autem  consequens  est,  ut  per- 


lat 


Which  standing,  it  follows  that  no  hindrance  can 
intervene. 

SCHOLION. 

Nevertheless  it  niight  be  doubted,  whether  a  perpen- 
dicular,  from  whatever  point  K  (assumed  indeed  in  BX 
before  the  meeting  of  this  BX  with  the  other  AX)  erected 
toward  the  parts  of  the  straight 
AX,  must  meet  this  (fig.  29)  in 
some  point  L;  provided  of  course 
those  two,  before  the  aforesaid 
meeting,  are  assumed  ever  more  to 
approach  each  other  mutually.  But 
I  say  it  will  follow  completely  thus. 

Proof.     Let  there  be  assigned 
in  BX  any  point  whatever  K.     In    ''^"         p.    ^g 
AX  is  taken  a  certain  AM  equal 
to  the  sum  of  this  BK  and  of  twice  AB. 

Then  from  the  point  M  is  drawn  to  BX  (according 
to  Eu.  L  12)  the  perpendicular  MN.  According  to  the 
present  supposition,  MN  will  be  less  than  AB.  Where- 
fore  AM  (made  equal  to  the  sum  of  BK  and  of  double 
AB)  will  be  greater  than  the  sum  of  BK,  AB,  and  NM. 
Now  it  behooves  to  show  this  same  AM  to  be  less  than  the 
sum  of  BN,  AB,  and  MN,  that  thence  it  may  follow  this 
BN  is  greater  than  the  aforesaid  BK,  and  therefore  the 
point  K  Hes  between  the  points  B  and  N. 

Join  BM.  The  side  AM  will  be  (from  Eu.  L  20)  less 
than  the  two  remaining  sides  together  AB  and  BM. 
Again  [51]  the  side  BM  (from  the  same  Eu.  I.  20)  will  be 
less  than  the  two  sides  together  BN  and  MN.  Therefore 
the  side  AM  will  be  by  much  less  than  the  three  sides  to- 
gether  AB,  BN,  and  NM.  But  this  was  to  be  shown,  in 
order  to  deduce  that  the  point  K  Hes  between  the  points 
B  and  N.    Thence  however  it  follows,  that  the  perpen- 


pendicularis  ex  puncto  K  educta  versus  partes  ipsius  AX 
occurrere  huic  debeat  in  aliquo  puncto  L  inter  puncta  A, 
et  M  constituto;  ne  scilicet  (contra  17.  primi)  secare  de- 
beat  alterutram  AB,  aut  MN  perpendiculares  eidem  BX. 
Quod  etc. 

PROPOSITIO  XXV. 

Si  duae  rectae  (fig.  30.)  AX,  BX  in  eodem  plano  exis- 
tentes  (una  quidem  sub  angulo  acuto  in  puncto  A, 
et  altera  in  puncto  B  perpendiculariter  insistens  ipsi 
AB)  ita  ad  se  invicem  semper  magis  accedant  versus 
partes  punctorum  X,  ut  nihilominus  earundem  dis- 
tantia  semper  major  sit  assignata  quadam  longitu- 
dine,  destruitur  hypothesis  anguli  acuti. 

Demonstratur.  Assignata  sit  longitudo  R.  Si  ergo  in 
ea  BX  sumatur  quaedam  BK  quantumlibet  multiplex  pro- 
positae  longitudinis  R;  constat  (ex  praecedente  Scholio) 
perpendicularem  ex  puncto  K  eductam  versus  partes  ip- 
sius  AX  in  aliquo  puncto  L  eidem  occursuram ;  ac  rursum 
(ex  praesente  hypothesi)  constat  eam  KL  majorem  fore 
praedicta  longitudine  R.  Porro  intelhgatur  BK  divisa  in 
portiones  KK,  aequales  singulas  ipsi  R,  usque  dum  KB 
aequahs  sit  ipsi  longitudini  R.  Tandem  vero  ex  punctis 
K  erectae  sint  ad  BX  perpendiculares  occurrentes  ipsi  AX 
in  punctis  L,  H,  D,  M,  usque  ad  punctum  N  proximius 
puncto  A.     Jam  sic  progredior. 

Erunt  (ex  Prop.  praecedente)  quatuor  simul  anguh 
quadrilateri  KHLK,  remotioris  ab  ea  basi  AB,  majores 


19» 


dicular  from  the  point  K  erected  toward  the  parts  of  AX 
must  meet  this  in  some  point  L  stationed  between  the 
points  A  and  M;  else  obviously  (against  Eu.  I.  17)  it 
must  cut  either  AB  or  MN  perpendiculars  to  BX. 
Quod  etc. 


PROPOSITION  XXV. 

//  two  straights  (fig.  30)  AX,  BX  existing  in  the  same 
plane  (standing  upon  AB,  one  indeed  at  an  acute 
angle  in  the  point  A,  and  the  other  perpendicular  at 
the  point  B)  so  always  approach  more  to  each  other 
mutually,  toward  the  parts  of 
the  point  X,  that  neverthe- 
less  their  distance  is  always 
greater  than  a  certain  as- 
signed  length,  the  hypothesis 
of  acute  angle  is  destroyed. 

Proof.  Let  R  be  the  assigned 
length.  If  therefore  in  BX  is  as- 
sumed  a  certain  BK  any  chosen 
multiple  of  the  proposed  length 
R;  it  follows  (from  the  preceding 
scholion)  that  the  perpendicular  erected  from  the  point  K 
toward  the  parts  of  AX  will  meet  it  at  some  point  L ;  and 
again  (from  the  present  hypothesis)  it  follows  that  this 
KL  will  be  greater  than  the  aforesaid  length  R.  Further- 
more  BK  is  understood  divided  into  portions  KK,  each 
equal  to  R,  even  until  KB  is  itself  equal  to  the  length  R. 
Finally  from  the  points  K  are  erected  to  BX  perpen- 
diculars  meeting  AX  in  points  L,  H,  D,  M,  even  to  the 
point  N  nearest  the  point  A. 

Now  I  proceed  thus. 

The  four  angles  together  of  the  quadrilateral  KHLK, 
more  remote  from  the  base  AB,  will  be  (from  the  pre- 


Fig.  30. 


131 


[52]  quatuor  simul  angulis  quadrilateri  KDHK,  proxi- 
mioris  eidem  basi ;  cujus  itidem  quadrilateri  quatuor  simul 
anguli  majores  erunt  quatuor  simul  angulis  subsequentis 
versus  eandem  basim  quadrilateri  KMDK.  Atque  ita 
semper  usque  ad  ultimum  quadrilaterum  KNAB,  cujus 
utique  quatuor  simul  anguli  minimi  erunt,  relate  ad  qua- 
tuor  simul  angulos  singulorum  ascendentium  versus  puncta 
X  quadrilaterorum. 

Quoniam  vero  tot  aderunt  praedicto  modo  recensita 
quadrilatera,  quot  sunt  praeter  basim  AB  demissae  ex 
punctis  ipsius  AX  ad  rectam  BX  perpendiculares ;  expen- 
denda  est  summa  omnium  simul  angulorum,  qui  compre- 
henduntur  in  illis  quadrilateris.  Ponamus  esse  novem 
ejusmodi  perpendiculares  demissas,  ac  propterea  novem 
itidem  quadrilatera.  Constat  (ex  13.  primi)  aequales 
esse  quatuor  rectis  angulos  hinc  inde  comprehensos  ad 
bina  puncta  illarum  octo  perpendicularium,  quae  mediae 
jaceant  inter  basim  AB,  et  remotiorem  perpendicularem 
LK.  Itaque  summa  horum  omnium  angulorum  erit  32 
rectorum.  Restant  duo  anguH  ad  perpendiculum  LK,  et 
duo  ad  basim  AB.  At  anguH,  unus  quidem  ad  punctum 
K,  et  alter  ad  punctum  B,  supponuntur  recti ;  angulus 
autem  ad  punctum  L  (ex  Cor.  post  XXIIL  hujus)  est 
obtusus.  Quapropter  (etiam  neglecto  angulo  acuto  ad 
punctum  A)  summa  omnium  angulorum,  qui  comprehen- 
duntur  ab  iUis  novem  quadrilateris,  excedet  35.  rectos. 
Inde  autem  fit,  ut  quatuor  simul  anguH  quadrilateri 
KHLK,  remotioris  a  basi,  minus  deficiant  a  quatuor  rec- 
tis,  quam  sit  nona  pars  unius  recti;  et  id  quidem  etiam 
si  aequahs  portio  praedicta  omnium  angulorum  summae 
contingeret  singuHs  ihis  quadrilateris.  Ergo  minor  adhuc 
erit  insinuatus  defectus,  cum  summa  quatuor  simul  angu- 


13Z 


ceding  proposition)  greater  [52]  than  the  four  angles  to- 
gether  of  the  quadrilateral  KDHK,  nearer  to  this  base; 
of  which  quadrilateral  in  the  same  way  the  four  angles 
together  will  be  greater  than  the  four  angles  together  of 
the  quadrilateral  KMDK  subsequent  toward  this  base. 
And  so  always  even  to  the  last  quadrilateral  KNAB, 
whose  four  angles  together  assuredly  will  be  the  least, 
in  reference  to  the  four  angles  together  of  each.  of  the 
quadrilaterals  ascending  toward  the  points  X. 

But  since  are  present  as  many  quadrilaterals  described 
in  the  aforesaid  manner,  as  are,  except  the  base  AB,  per- 
pendiculars  let  fall  from  points  of  AX  to  the  straight 
BX;  the  sum  of  all  the  angles  together,  which  are  com- 
prehended  in  these  quadrilaterals  can  be  reckoned.  We 
assume  that  there  are  nine  such  perpendiculars  let  fall, 
and  therefore  so  nine  quadrilaterals. 

We  get  (from  Eu.  I.  13)  as  equal  to  four  rights  the 
angles  comprehended  hither  and  yon  at  the  two  points  of 
those  eight  perpendiculars,  which  He  in  the  middle  be- 
tween  the  base  AB  and  the  more  remote  perpendicular 
LK.     So  the  sum  of  all  these  angles  will  be  32  rights. 

There  remain  two  angles  at  the  perpendicular  LK, 
and  two  at  the  base  AB.  But  the  angles  one  indeed  at  the 
point  K  and  the  other  at  the  point  B  are  supposed  right ; 
but  the  angle  at  the  point  L  (from  the  Cor.  to  P.  XXIIL) 
is  obtuse.  Wherefore  (even  neglecting  the  acute  angle  at 
the  point  A)  the  sum  of  all  the  angles  which  are  compre- 
hended  by  these  nine  quadrilaterals  exceeds  35  rights. 
But  hence  follows,  that  the  four  angles  together  of  the 
quadrilateral  KHLK,  more  remote  from  the  base  lack 
less  from  four  rights  than  the  ninth  part  of  one  right; 
and  that  indeed  even  if  an  equal  portion  of  the  aforesaid 
sum  of  all  the  angles  pertained  to  each  of  those  quad- 
rilaterals. 

Therefore  less  yet  will  be  the  occurring  defect,  since 

133 


lorum  illius  quadrilateri  KHLK  ostensa  sit  omnium 
maxima,  relate  ad  [53]  quatuor  simul  angulos  reliquorum 
quadrilaterorum. 

Sed  rursum;  juxta  suppositionem,  in  qua  procedit 
haec  Propositio ;  assumi  potest  tanta  longitudo  ipsius  BK, 
ut  confici  semper  possint  non  tot  quin  plura  quadrilatera 
sub  basibus  KK,  aequalibus  singulis  illi  assignatae  longi- 
tudini  R.  Quare  defectus  quatuor  simul  angulorum  illius 
remotioris  quadrilateri  KHLK  a  quatuor  rectis  osten- 
detur  semper  minor  et  una  centesima,  et  una  millesima,  et 
sic  sub  quolibet  assignabili  numero  una  portiuncula  unius 
recti. 

Porro  autem  erunt  semper  (juxta  praedictam  suppo- 
sitionem)  ipsa  LK,  et  HK  majores  designata  longitudine 
R.  Si  ergo  in  KL,  et  KH  sumantur  KS,  et  KT  aequales 
ipsi  KK,  seu  longitudini  R;  erunt,  juncta  ST,  duo  simul 
anguli  KST,  KTS  majores,  in  hypothesi  anguli  acuti, 
duobus  simul  anguHs  (ex  Cor.  post  XVL  hujus)  ad 
puncta  H,  et  L  in  quadrilatero  THLS,  seu  quadrilatero 
KHLK;  ac  propterea  (additis  communibus  rectis  angulis 
ad  puncta  K,  K)  erunt  quatuor  simul  anguH  quadrilateri 
KTSK  majores  quatuor  simul  anguHs  ihius  quadrilateri 
KHLK. 

Jam  vero:  cum  ex  una  parte  stabile  sit,  ac  datum 
quadrilaterum  KTSK,  utpote  constans  data  basi  KK, 
quae  nimirum  aequaHs  ponitur  assignatae  longitudini  R, 
ac  rursum  constans  duobus  perpendicuHs  TK,  SK  eidem 
basi  aequaHbus,  ac  tandem  jungente  TS,  quae  evadit 
omnino  determinata;  et  ex  altera  quatuor  simul  anguH 
stabiHs  iUius,  ac  dati  quadrilateri,  ostensi  jam  sint  majo- 
res  quatuor  simul  anguHs  quadrilateri  KHLK  quantum- 
Hbet  distantis  ab  ea  basi  AB:  consequens  utique  fit,  ut 
quatuor  simul  anguH  stabiHs  iUius,  ac  dati  quadrilateri 


134 


the  sum  of  the  four  angles  together  of  this  quadrilateral 
KHLK  was  shown  the  greatest  of  all,  in  relation  to  [53] 
the  four  angles  together  of  the  remaining  quadrilaterals. 

But  again;  in  consequence  of  the  supposition  upon 
which  this  proposition  proceeds,  so  great  a  length  of  BK 
can  be  assumed,  that  as  many  quadrilaterals  as  we  choose 
may  be  made  on  bases  KK,  each  equal  to  the  assigned 
length  R. 

Wherefore  the  defect  of  the  four  angles  together  of 
this  more  remote  quadrilateral  KHLK  from  four  rights 
is  shown  ever  less  both  than  a  hundredth  and  than  a 
thousandth,  and  thus  under  any  assignable  part  of  a 
right.  Further  however,  LK  and  HK  will  be  always  (in 
accordance  with  the  aforesaid  supposition)  greater  than 
the  designated  length  R.  Therefore  if  in  KL  and  KH 
are  assumed  KS  and  KT  equal  to  KK  or  the  length  R; 
ST  being  joined,  the  two  angles  together  KST,  KTS 
will  be  greater,  in  hypothesis  of  acute  angle,  than  the  two 
angles  together  (from  Cor.  to  P.  XVL)  at  the  points  H 
and  L  in  the  quadrilateral  THLS,  or  the  quadrilateral 
KHLK;  and  therefore  (the  common  right  angles  at  the 
points  K,  K  being  added)  the  four  angles  together  of 
the  quadrilateral  KTSK  will  be  greater  than  the  four 
angles  together  of  that  quadrilateral  KHLK. 

But  now,  since  on  one  hand  is  stable  and  given  the 
quadrilateral  KTSK,  inasmuch  as  constant  in  the  given 
base  KK,  which  indeed  is  taken  equal  to  the  assigned 
length  R,  and  again  constant  in  the  two  perpendiculars 
TK,  SK  equal  to  this  base,  and  finally  in  the  joining  TS, 
which  comes  out  completely  determinate;  and  on  the 
other  hand  the  four  angles  together  of  this  stable  and 
given  quadrilateral  have  now  been  shown  greater  than  the 
four  angles  together  of  the  quadrilateral  KHLK  distant 
as  far  as  we  choose  from  the  base  AB;  assuredly  it  fol- 
lows  that  the  four  angles  together  of  this  stable  and  given 

135 


KTSK  majores  sint  qualibet  angulorum  summa,  quae 
quomodolibet  deficiat  a  quatuor  rectis;  quandoquidem 
ostensum  jam  est  designari  semper  posse  tale  aliquod 
quadrilaterum  KHLK,  [54]  cujus  quatuor  simul  anguli 
minus  deficiant  a  quatuor  rectis,  quam  sit  quaevis  desig- 
nabilis  unius  recti  portiuncula.  Igitur  quatuor  simul 
anguli  stabilis  illius,  ac  dati  quadrilateri,  vel  aequales  sunt 
quatuor  rectis,  vel  eisdem  majores.  Tunc  autem  (ex 
XVI.  hujus)  stabilitur  hypothesis  aut  anguH  recti,  aut 
anguH  obtusi;  ac  propterea  (ex  V.  et  VI.  hujus)  destrui- 
tur  hypothesis  anguH  acuti. 

Itaque  constat  destructum  iri  hypothesim  anguH  acuti, 
si  duae  rectae  in  eodem  plano  existentes  ita  ad  se  invicem 
semper  magis  accedant,  ut  nihilominus  earundem  distan- 
tia  major  semper  sit  assignata  quadam  longitudine.  Hoc 
autem  erat  demonstrandum. 

COROLLARIUM  I. 

At  (destructa  hypothesi  anguH  acuti)  manifestum  fit, 
ex  13.  hujus,  controversum  Pronunciatum  EucHdaeum; 
prout  a  me  hoc  loco  declaratum  iri  spopondi  in  SchoHo 
III.  post  XXI.  hujus,  ubi  de  conatu  Nassaradini  Arabis 
locuti  sumus. 

COROLLARIUM  11. 

Rursum  ex  hac  Propositione,  et  ex  praecedente 
XXIII.  manifeste  coHigitur  satis  non  esse  ad  stabiHendam 
Geometriam  EucHdaeam  duo  puncta  sequentia.  Unum 
est:  quod  nomine  paraHelarum  ihas  rectas  censeamus, 
quae  in  eodem  plano  existentes  commune  aHquod  obtinent 
perpendiculum.  Alterum  vero,  quod  omnes  rectae  in 
eodem  plano  existentes,  quarum  nuHum  commune  sit  per- 


136 


quadrilateral  KTSK  are  greater  than  any  sum  of  angles, 
which  lacks  however  Httle  you  choose  of  being  four  right 
angles;  since  already  it  has  been  shown  that  a  quadri- 
lateral  KHLK  can  always  be  designated  such  [54]  that 
its  four  angles  together  shall  fall  short  of  four  rights 
by  less  than  any  assignable  part  of  a  right.  Therefore 
the  four  angles  together  of  this  stable  and  given  quadri- 
lateral  either  are  equal  to  four  rights  or  greater. 

But  then  (from  P.  XVI.)  is  estabHshed  the  hypoth- 
esis  either  of  right  angle  or  of  obtuse  angle;  and  there- 
fore  (from  Propp.  V.  and  VI.)  the  hypothesis  of  acute 
angle  is  destroyed. 

So  is  estabHshed  that  the  hypothesis  of  acute  angle 
will  be  destroyed,  if  two  straights  existing  in  the  same 
plane  so  approach  each  other  mutuaUy  ever  more,  that 
nevertheless  their  distance  is  always  greater  than  any 
assigned  length. 

Hoc  autem  erat  demonstrandum. 

COROLLARY  L 

But  (the  hypothesis  of  acute  angle  destroyed)  the 
controverted  EucHdean  postulate  is  manifest  from  P. 
XIII. ;  just  as  in  SchoHon  III.  after  P.  XXL,  where  we 
spoke  of  the  attempt  of  the  Arab  Nasiraddin,  I  promised 
would  be  disclosed  by  me  in  this  place. 

COROLLARY  IL 

On  the  other  hand  from  this  proposition,  and  from 
the  preceding  P.  XXIII.  is  manifestly  gathered  that  the 
two  foHowing  points  are  not  sufficient  for  estabHshing 
EucHdean  geometry.  One  is :  that  we  designate  by  the 
name  of  paraHels  those  straights,  which  existing  in  the 
same  plane  possess  a  common  perpendicular.  The  second 
indeed,  that  all  straights  existing  in  the  same  plane,  of 


137 


pendiculum,  ac  propterea  quae  juxta  assumptam  Defini- 
tionem  parallelae  non  sint,  debeant  ipsae  in  alterutram 
partem  semper  magis  protractae  inter  se  aliquando  inci- 
dere,  si  non  ad  [55]  finitam,  saltem  ad  infinitam  distan- 
tiam.  Nam  rursum  demonstrare  oporteret,  quod  duae 
quaelibet  in  eodem  plano  existentes,  in  quas  recta  quae- 
piam  incidens  duos  ad  easdem  partes  internos  angulos 
efiiciat  minores  duobus  rectis,  nusquam  alibi  possint  ipsae 
recipere  commune  perpendiculum.  Quod  autem,  hoc 
demonstrato,  exactissime  stabiliatur  Geometria  Euclidaea, 
infra  constabit. 

PROPOSITIO  XXVI. 

Si  praedictae  AX,  BX  {fig.  31.)  coire  quidem  inter  se 
deheant,  sed  non  nisi  ad  infinitam  earundem  produc- 
tionem  versus  partes  punctorum  X:  Dico  nullum 
fore  assignahile  punctum  I  in  ipsa  AB,  ex  quo  per- 
pendicularis  educta  versus  partes  ipsius  AX  non 
occurrat  ad  finitam,  seu  terminatam  distantiam  eidem 
AX  in  aliquo  puncto  F. 

Demonstratur.      Nam    (ex    praecedente    hypothesi) 
unum  aliquod  erit  in  ipso  AX  punctum  N,  ex  quo  per- 


iji 


which  there  is  no  common  perpendicular,  and  therefore 
which  according  to  the  assumed  definition  are  not  parallel; 
must,  being  produced  toward  either  part  ever  more,  some- 
where  meet  each  other,  if  not  at  [55]  a  finite,  at  least  at 
an  infinite  distance. 

For  again  it  would  be  requisite  to  demonstrate,  that 
any  two  straights  existing  in  the  same  plane,  upon  which 
a  certain  straight  cutting  makes  two  internal  angles  toward 
the  same  parts  less  than  two  right  angles,  nowhere  else 
can  receive  a  common  perpendicular. 

But  that,  this  demonstrated,  EucHdean  geometry  is 
most  exactly  estabhshed,  will  be  shown  below. 

PROPOSITION  XXVI. 

//  the  aforesaid  AX,  BX  (fig.  31)  must  indeed  meet 
each  other,  but  only  at  their  infinite  production 
toward  the  parts  of  the  point  X:  I  say  there  zvill 


he  no  assignable  point  T  in  AB,  from  which  a  per- 
pendicular  erected  toward  the  parts  of  AX  does  not 
at  a  finite,  or  terminated  distance  meet  this  AX  in 
some  point  F. 

Proof.     For  (from  the  preceding  hypothesis)  there 
will  be  in  AX  some  point  N,  from  which  the  perpen- 

139 


pendicularis  NK  demissa  ad  BX  minor  sit  qualibet  assig- 
nata  longitudine,  ut  puta  ea  TB.  Tum  vero  sumatur  in 
TB  portio  CB  aequalis  ipsi  NK,  jungaturque  CN.  Con- 
stat  angulum  NCB  acutum  fore,  in  hypothesi  anguH  acuti. 
Ergo  (ex  13.  primi)  obtusus  erit,  qui  deinceps  est  angulus 
NCT.  Igitur  recta,  quae  ex  puncto  T  (inter  puncta  A,  et 
C  constituto)  perpendiculariter  educatur  versus  partes 
ipsius  AX,  non  incidet  (ex  17.  primi)  in  uUum  punctum 
ipsius  CN;  ac  propterea  (ne  claudat  spatium  cum  AT, 
aut  cumTC)  occurret  ipsi  terminatae  AN  in  aHquo  puncto 
F.  Igitur  in  ipsa  etiam  hypothesi  anguH  acuti  (quam 
scimus  obesse  unice  hic  posse)  nuHum  erit  assignabile 
punctum  T  in  ea  AB,  ex  quo  perpendiculariter  educta 
versus  partes  ipsius  AX  non  occurrat  ad  finitam,  seu  ter- 
minatam  distantiam  eidem  AX  in  quodam  puncto  F. 
Quod  etc.  [56] 

COROLLARIUM  L 

Inde  autem  fit,  ut  assumpto  in  AB  protracta  quoHbet 
puncto  M,  ex  quo  versus  partes  punctorum  X  educatur 
perpendicularis  MZ,  nequeat  ipsa,  etiamsi  infinite  produ- 
catur,  occurrere  praedictae  AX ;  quia  caeterum  iUa  altera 
BX  deberet  (ex  praemissa  demonstratione)  ad  finitam 
distantiam  occurrere  eidem  AX ;  quod  est  contra  praesen- 
tem  hypothesin. 

COROLLARIUM  11. 

Ex  quo  rursum  consequitur  omnem  perpendiculariter 
eductam  ex  quoHbet  puncto  iUius  quantumHbet  continuatae 
AB,  sed  non  tamen  infinite  dissito,  debere  ad  finitam  dis- 
tantiam  occurrere  praedictae  AX;  quatenus  nempe  sup- 
ponatur  omnem  talem  perpendiculariter  eductam  semper 


140 


dicular  NK  let  fall  to  BX  is  less  than  any  assigned  length, 
as  suppose  than  TB.  But  then  is  assumed  in  TB  a  por- 
tion  CB  equal  to  NK  and  CN  is  joined.  In  the  hypothesis 
of  acute  angle,  it  is  known  that  the  angle  NCB  will  be 
acute.  Therefore  (from  Eu.  I.  13)  NCT,  which  is  the 
adjacent  angle,  will  be  obtuse. 

Therefore  the  straight,  which  is  erected  toward  the 
parts  of  AX  perpendicularly  from  the  point  T  (disposed 
between  the  points  A  and  C),  does  not  meet  (from  Eu. 
I.  17)  CN  at  any  point;  and  therefore  (lest  it  should 
inclose  a  space  with  AT,  or  with  TC)  it  strikes  the  ter- 
minated  AN  in  some  point  F. 

Therefore  even  in  the  hypothesis  of  acute  angle  (which 
we  know  can  here  alone  hinder)  there  will  be  in  this 
AB  no  assignable  point  T,  from  which  the  perpendicular 
erected  toward  the  parts  of  AX  does  not,  at  a  finite  or 
terminated  distance,  meet  this  AX  in  a  certain  point  F. 

Quod  erat  etc.  [56] 

COROLLARY  1. 

But  thence  follows,  that,  point  M  being  assumed  in 
AB  produced,  from  which  is  erected  toward  the  parts  of 
the  points  X  a  perpendicular  MZ,  this  cannot,  even  if 
infinitely  produced,  meet  the  af  oresaid  AX ;  because  other- 
wise  that  other  straight  BX  must  (from  the  foregoing 
demonstration)  at  a  finite  distance  meet  this  AX;  which 
is  against  the  present  hypothesis. 

COROLLARY  IL 

From  which  again  it  follows,  that  every  perpendicular, 
erected  from  any  point  (but  not  however  infinitely  re- 
moved)  of  this  AB  produced  indefinitely,  must  at  a 
finite  distance  meet  the  aforesaid  AX,  as  soon  as  indeed 
it  is  assumed  that  every  such  perpendicular  ever  more, 

141 


magis,  sine  ullo  certo  limite  accedere  ad  alteram  semper 
continuatam  AX.  ^^~ 

COROLLARIUM  IIL 

Unde  tandem  fit,  ut  ab  illa  AX  neque  ad  infinitam 
ejusdem  productionem  secari  possit  ipsa  BX;  quia  caete- 
rum  ex  quodam  illius  AX  ultra  praedictam  sectionem 
puncto  intelligi  posset  demissa  ad  AB  productam  quae- 
dam  perpendicularis  ZM ;  unde  rursum  fieret,  ut  ipsa  BX 
(contra  praesentem  hypothesim)  non  ad  infinitam,  sed 
omnino  ad  finitam  distantiam  occurreret  praedictae  AX. 
Sed  hoc  postremum  dictum  sit  ultra  necessitatem.  [57] 

PROPOSITIO  XXVII. 

Si  recta  AX  (fig.  32.)  sub  aliquo,  ut  lihet,  parvo  angulo 
educta  ex  puncto  A  ipsius  AB,  occurrere  tandem  de- 
beat  (saltem  ad  infinitam  distantiam)  cuivis  perpen- 
diculari  BX,  quae  ad  quandamlibet  ab  eo  puncto  A 
distantiam  excitari  intelligatur  super  ea  incidente 
AB:  Dico  nullum  jam  fore  locum  hypothesi  anguli 
acuti. 

Demonstratur.  Ex  quodam  puncto  K  prope  punctum 
A,  ad  libitum  in  ipsa  AB  designato,  erigatur  ad  AB  per- 
pendicularis  KL,  quae  utique  (ex  Cor.  11.  praecedentis 
Propositionis)  occurret  ipsi  AX  ad  finitam,  seu  termina- 


J4* 


without  any  certain  limit,  approaches  the  other  ever  pro- 
duced  straight  AX. 


COROLLARY  IIL 

Whence  finally  follows,  that  not  even  at  its  infinite 
production  can  BX  be  cut  by  that  AX ;  because  otherwise 
from  any  point  of  that  AX  beyond  the  aforesaid  inter- 
section  a  certain  perpendicular  ZM  could  be  supposed 
let  f all  to  AB  produced ;  whence  again  would  f ollow,  that 
BX  (against  the  present  hypothesis)  met  the  aforesaid 
AX  not  at  an  infinite,  but  wholly  at  a  finite  distance. 

But  this  last  dictum  is  beyond  necessity.  [57] 

PROPOSITION  XXVIL 

//  a  straight  AX  (fig.  32)  drawn  at  any  hoivever  small 
angle  from  the  point  A  of  AB,  must  at  length  meet 
(anyhow  at  an  infinite  distance)  any  perpendicular 
BX,  which  is  supposed  erected  at  any  distance  from 
this  point  A  upon  the  secant  AB :  /  say  there  will 
then  be  no  more  place  for  the  hypothesis  of  acute 
angle. 


Proof.  From  any  point  K  chosen  at  will  in  AB  near 
the  point  A,  the  perpendicular  KL  is  erected  to  AB, 
which  certainly  (from  Cor.  II.  of  the  preceding  propo- 
sition)  meets  AX  at  a  finite  or  terminated  distance  in 


143 


tam  distantiam  in  aliquo  puncto  L.  Jam  vero  constat  sumi 
posse  in  KB  portiones  KK  aequales  singulas  cuidem  as- 
signabili  longitudini  R,  et  eas  plures  quolibet  assignabili 
numero  finito;  quandoquidem  punctum  B  statui  potest; 
juxta  praesentem  suppositionem ;  in  quantalibet  distantia 
ab  eo  puncto  A.  Itaque  ex  aliis  punctis  K  erigantur  ad 
AB  perpendiculares  KH,  KD,  KP,  quae  omnes  (ex  prae- 
citato  Corollario)  occurrent  rectae  AX  in  quibusdam 
punctis  H,  D,  P;  atque  ita  circa  reliqua  puncta  K  uni- 
formiter  designata  versus  punctum  B.  Constat  secundo 
(ex  16.  primi)  angulos  ad  puncta  L,  H,  D,  P,  fore  omnes 
obtusos  versus  partes  punctorum  X;  atque  item  (ex  13. 
ejusdem  primi)  angulos  ad  praedicta  puncta  fore  omnes 
acutos  versus  punctum  A.  Igitur  (ex  Cor.  II.  post  3. 
hujus)  latus  KH  majus  erit  latere  KL;  latus  KD  majus 
latere  KH;  atque  ita  semper,  procedendo  versus  puncta 
X.  Constat  tertio  quatuor  simul  angulos  quadrilateri 
KLHK  majores  fore  quatuor  simul  angulis  quadrilateri 
KHDK:  nam  id  in  simili  demonstratum  jam  est  in 
XXIV.  hujus.  Constat  quarto  idem  similiter  valere  de 
quadrilatero  KHDK  relate  ad  quadrilaterum  KDPK; 
atque  ita  semper,  procedendo  ad  qua-[58]drilatera  remo- 
tiora  ab  eo  puncto  A. 

Quoniam  igitur  tot  aderunt  (ut  in  XXV.  hujus) 
praedicto  modo  recensita  quadrilatera,  quot  sunt,  praeter 
primam  LK,  demissae  ex  punctis  ipsius  AX  perpendicu- 
lares  ad  rectam  AB;  constabit  uniformiter  (si  ponamus 
novem,  praeter  primam,  demissas  ejusmodi  perpendicu- 
lares)  summam  omnium  angulorum,  qui  comprehendun- 
tur  ab  illis  novem  quadrilateris,  excedere  35.  rectos;  ac 


H* 


some  point  L.  But  now  it  holds  that  there  may  be  as- 
sumed  in  KB  portions  KK  each  equal  to  a  certain  assign- 
able  length  R,  and  these  more  than  any  assignable  finite 
number;  since  indeed  the  point  B  can  be  situated,  in 
accordance  with  the  present  supposition,  at  however  great 
a  distance  from  this  point  A. 

And  accordingly  from  the  other  points  K  are  erected 
to  AB  perpendiculars  KH,  KD,  KP,  which  all  (from 
the  aforesaid  corollary)  meet  the  straight  AX  in  certain 
points  H,  D,  P;  and  so  about  the  remaining  points  K 
uniformly  designated  toward  the  point  B. 

It  holds  secondly  (from  Eu.  I.  16)  that  the  angles 
at  the  points  L,  H,  D,  P  will  all  be  obtuse  toward  the 
parts  of  the  points  X;  and  just  so  (from  Eu.  L  13)  the 
angles  at  the  aforesaid  points  will  all  be  acute  toward 
the  point  A. 

Therefore  (from  Cor.  H.  to  P.  HL)  the  side  KH 
will  be  greater  than  the  side  KL;  the  side  KD  greater 
than  the  side  KH ;  and  so  always  proceeding  toward  the 
points  X. 

It  holds  thirdly  that  the  four  angles  together  of  the 
quadrilateral  KLHK  will  be  greater  than  the  four  angles 
together  of  the  quadrilateral  KHDK :  for  this  in  like  case 
has  already  been  demonstrated  in  P.  XXIV. 

It  holds  fourthly  that  the  same  is  valid  likewise  of 
the  quadrilateral  KHDK  in  relation  to  the  quadrilateral 
KDPK;  and  so  on  always,  proceeding  to  quadrilaterals 
[58]  more  remote  f  roni  this  point  A. 

Since  therefore  are  present  (as  in  P.  XXV.)  as  many 
quadrilaterals  described  in  the  aforesaid  mode,  as  there 
are,  except  the  first  LK,  perpendiculars  let  fall  f rom  points 
of  AX  to  the  straight  AB,  it  will  hold  uniformly  (if  we 
assume  nine  perpendiculars  of  this  sort  let  fall,  besides 
the  first)  the  sum  of  all  the  angles  which  are  compre- 
hended  by  these  nine  quadrilaterals  will  exceed  35  right 

M5 


propterea  quatuor  simul  angulos  primi  quadrilateri 
KLHK,  quod  quidem  in  hac  ratione  ostensum  est  om- 
nium  maximum,  minus  deficere  a  quatuor  rectis,  quam 
sit  nona  pars  unius  recti.  Quare;  multiplicatis  ultra 
quemlibet  assignabilem  finitum  numerum  eisdem  quadri- 
lateris,  procedendo  semper  versus  partes  punctorum  X; 
constabit  similiter  (ut  in  eadem  praecitata)  quatuor  simul 
angulos  stabilis  illius  quadrilateri  KHLK  minus  deficere 
a  quatuor  rectis,  quam  sit  quaelibet  assignabilis  unius  recti 
portiuncula.  Igitur  quatuor  simul  illi  anguli  vel  aequales 
erunt  quatuor  rectis,  vel  eisdem  majores.  Tunc  autem 
(ex  XVL  hujus)  stabilitur  hypothesis  aut  anguli  recti, 
aut  anguH  obtusi;  ac  propterea  (ex  V.  et  VL  hujus)  de- 
struitur  hypothesis  anguli  acuti. 

Itaque  constat  nullum  jam  fore  locum  hypothesi  an- 
guli  acuti,  si  recta  AX  sub  aliquo,  ut  libet,  parvo  angulo, 
educta  ex  puncto  A  ipsius  AB  occurrere  tandem  debeat 
(saltem  ad  infinitam  distantiam)  cuivis  perpendiculari 
BX,  quae  ad  quantamlibet  ab  eo  puncto  A  distantiam  ex- 
citari  intelligatur  super  ea  incidente  AB.    Quod  erat  etc. 

SCHOLION  I. 

Et  hoc  est,  quod  praedixi  in  Cor.  II.  post  XXV.  hujus ; 
nullum  scilicet  superfuturum  locum  hypothesi  an-[59]guli 
acuti,  seu  stabilitum  exactissime  iri  Geometriam  Eucli- 
daeam ;  si  duae  quaelibet  in  eodem  plano  existentes  rectae, 
ut  puta  AX,  BX,  in  quas  incidens  recta  AB  (sumpto 
puncto  B  in  quantalibet  distantia  a  puncto  A)  duos  cum 
eisdem  ad  easdem  partes  punctorum  X  angulos  efficiat 
minores  duobus  rectis;  si  (inquam)  nusquam  alibi  (hoc 


146 


angles;  and  therefore  the  four  angles  together  of  the  first 
quadrilateral  KLHK,  which  indeed  in  this  regard  has 
been  shown  the  greatest  of  all,  will  fall  short  of  four 
right  angles  by  less  than  the  ninth  part  of  one  right  angle. 
Wherefore,  these  quadrilaterals  being  multiplied  beyond 
any  assignable  finite  number,  proceeding  always  toward 
the  parts  of  the  points  X,  it  holds  in  the  same  way  (as  in 
the  same  already  recited  theorem)  that  the  four  angles 
together  of  this  stable  quadrilateral  KHLK  will  f  all  short 
of  four  right  angles  less  than  any  assignable  little  portion 
of  one  right  angle. 

Therefore  these  four  angles  together  will  be  either 
equal  to  four  right  angles,  or  greater. 

But  then  (from  P.  XVL)  is  established  the  hypoth- 
esis  of  right  angle  or  of  obtuse  angle;  and  therefore 
(from  Propp.  V.  and  VL)  is  destroyed  the  hypothesis 
of  acute  angle. 

So  then  it  holds,  that  there  will  be  no  place  for  the 
hypothesis  of  acute  angle,  if  the  straight  AX  drawn  under 
however  small  angle  from  the  point  A  of  AB  must  at 
length  meet  (anyhow  at  an  infinite  distance)  any  perpen- 
dicular  BX,  which  is  supposed  erected  at  any  distance 
from  this  point  A  upon  this  secant  AB. 

Quod  erat  etc. 

SCHOLION  I. 

And  this  it  is,  that  I  said  before  in  Cor.  H.  to  P. 
XXV.;  obviously  that  no  place  would  remain  over  for 
the  hypothesis  of  acute  angle,  [59]  or  Euclidean  geometry 
would  be  most  exactly  established,  if  any  two  straights 
existing  in  the  same  plane,  as  suppose  AX,  BX,  which 
the  straight  AB  meeting  (the  point  B  being  assumed  at 
a  distance  from  the  point  A  as  great  as  you  choose) 
makes  with  them  toward  the  same  parts  of  the  points  X 
two  angles  less  than  two  right  angles,  if  (I  say)  nowhere 

147 


stante)  possint  illae  recipere  commune  perpendiculum. 
Tunc  enim  illae  duae  AX,  BX,  semper  magis  ad  se  in- 
vicem  accedent;  nimirum  vel  intra  quendam  determi- 
natum  limitem,  prout  in  XXV.  hujus;  vel  sine  ullo  certo 
limite,  ac  propterea  usque  ad  occursum  saltem  post  infini- 
tam  productionem,  prout  in  hac  XXVII.  Constat  autem 
in  utroque  praedictorum  casuum  ostensam  jam  esse  de- 
structionem  hypothesis  anguH  acuti.    Quod  intendebatur. 

SCHOLION  II. 

Atque  id  rursum  est,  quod  spopondi  in  fine  Scholii 
IV.  post  XXI.  hujus,  prout  ex  ipsis  terminis  clare  elu- 
cescit. 

SCHOLION  III. 

Praeterea  observari  hic  veHm  discrimen  inter  hanc 
Propos.  et  praecedentem  XVII.  Nam  ibi  (recole  fig.  15.) 
ostensa  est  destructio  hypothesis  anguH  acuti,  si  (existente, 
ut  Hbet  parva,  recta  AB)  omnis  BD  sub  quovis  acuto 
angulo  educta,  occurrere  tandem  debeat  in  quodam  puncto 
K  ipsi  perpendiculari  AH  productae.  Hic  autem  (vice- 
versa)  permittitur  quidem  designatio  cujusvis  parvissimi 
acuti  anguH  ad  punctum  A,  dum  tamen  interjecta  AB, 
ad  quam  erigenda  est  perpendicularis  indefinita  [60]  BX, 
statui  possit  quantaeHbet  longitudinis. 

PROPOSITIO  xxvin. 

Si  duae  rectae  AX^  BX  (quarum  prior  sub  angulo  acuto, 
et  altera  ad  perpendiculum  eductae  sint  versus  eas- 
dem  partes  ex  quantalibet  recta  AB)  semper  magis 


14« 


at  another  place  (this  standing)  they  can  admit  a  com- 
mon  perpendicular. 

For  then  these  two  AX,  BX  mutually  approach  each 
other  ever  more,  indeed  either  within  a  certain  deter- 
minate  Hmit,  as  in  P.  XXV.,  or  without  any  certain  limit, 
and  therefore  even  to  meeting,  anyhow  after  infinite  pro- 
duction,  as  in  P.  XXVII. 

But  it  holds  that  in  either  of  the  aforesaid  cases  the 
destruction  of  the  hypothesis  of  acute  angle  has  now  been 
shown. 

Quod  intendebatur. 

SCHOLION  II. 

And  again  this  it  is,  that  I  promised  at  the  end  of 
Schohon  IV.  after  P.  XXI.,  as  from  the  very  terms 
clearly  appears. 

SCHOLION  III. 

Moreover  I  could  wish  here  to  be  observed  the  differ- 
ence  between  this  proposition  and  the  preceding  P.  XVII. 
For  there  (recall  fig.  15)  has  been  shown  the  destruction 
of  the  hypothesis  of  acute  angle,  if  (the  straight  AB  being 
as  small  as  you  choose)  every  BD  erected  at  whatever 
acute  angle,  must  at  length  meet  in  some  point  K  the  per- 
pendicular  AH  produced. 

But  here  (vice  versa)  in  fact  is  permitted  the  desig- 
nation  of  however  most  small  an  acute  angle  at  the  point 
A  while  still  the  sect  AB  to  which  is  to  be  erected  the  in- 
definite  perpendicular  [60]  BX,  may  be  taken  of  any  length 
whatever. 

PROPOSITION  XXVIII. 

//  two  straights  AX,  BX  (produced  front  any-sized 
straight  AB  toward  the  same  parts,  the  first  under 
an  acute  angle,  and  the  other  perpendicularly)  mu- 


sine  ullo  certo  limite  ad  se  invicem  accedant,  praeter- 
quam  ad  infinitam  earundem  productionem;  Dico 
omnes  angulos  (fig.  33.)  ad  quaelibet  puncta  L,  H,  D 
ipsius  AX,  ex  quibus  demittantur  ad  rectam  BX  per- 
pendiculares  LK,  HK,  DK;  tum  fore  omnes  obtusos 
versus  partes  puncti  A;  tum  fore  semper  minores,  qui 
magis  distant  ab  eo  puncto  A;  ac  tandem  angulos 
magis,  ac  magis  distantes  ab  eodem  puncto  A,  semper 
magis  sine  ullo  certo  limite  accedere  ad  aequalitatem 
cum  angulo  recto. 

Demonstratur.  Et  prima  quidem  pars  constat  ex 
Cor.  I.  post  XXIII.  hujus.  Secunda  vero  pars  ita  evin- 
citur.  Nam  duo  simul  anguli  ad  LK  versus  basim  AB 
majores  sunt  (ex  Cor.  post  XVI.  hujus)  duobus  simul 
internis,  et  oppositis  anguhs  ad  HK  versus  eandem  basim 
AB.  Sunt  autem  inter  se  aequales,  utpote  recti,  anguH 
ad  utrunque  punctum  K  versus  basim  AB.  Ergo  angulus 
obtusus  ad  L  versus  basim  AB  major  est  angulo  obtuso 
ad  H  versus  eandem  basim  AB.  Simih  modo  ostendetur 
praedictum  angulum  obtusum  ad  H  majorem  esse  angulo 
obtuso  ad  punctum  D.  Atque  ita  semper,  procedendo 
versus  puncta  X. 

Tertia  tandem  pars  majore  indiget  disquisitione.  Si 
ergo  fieri  potest,  assignatus  sit  (fig.  34.)  quidam  angulus 
MNC,  quo  semper  major  sit,  aut  saltem  non  minor,  ex- 
cessus  cujusvis  ex  praedictis  anguhs  obtusis  supra  angu- 
lum  rectum.  Constat  (ex  XXI.  hujus)  latera  NM,  NC 
comprehendentia  ihum  angulum  MNC  tahter  produci 
posse,  ut  perpendicularis  MC,  ex  quodam  puncto  M  ipsius 
MN  [61]  demissa  ad  NC,  major  sit  (in  ipsa  etiam  hypo- 


is» 


tually  approach  each  other  ever  more  without  any 
certain  limit,  save  at  their  infinite  production ;  /  say 
all  angles  (fig.  33)  at  any  points  L,  H,  D  of  AX, 
from  which  are  let  fall  to  the  straight  BX  perpen- 
diculars  LK,  HK,  DK,  first  will  all  be  obtuse  toward 
the  parts  of  the  point  A,  secondly  will  be  ever  less, 
the  more  distant  from  this  point  A,  and  finally  the 
angles  more  and  more  distant  from  this  same  point 
A  ever  more  without  any  certain  limit  approach  to 
equality  with  a  right  angle. 

Proof.  The  first  part  follows  indeed  f rom  Cor.  I.  to 
P.  XXIII.  The  second  part  however  is  proved  thus.  For 
the  two  angles  together  at  LK  toward 
the  base  AB  are  greater  (from  Cor. 
to  P.  XVI.)  than  the  two  internal 
and  opposite  angles  together  at  HK 
toward  the  same  base  AB. 

But  the  angles  at  each  point  K 
toward  the  base  AB  are  equal  to  each 
other,  as  being  right.  Therefore  the 
obtuse  angle  at  L  toward  the  base 
AB  is  greater  than  the  obtuse  angle  '^  Fig.  ZZ. 
at  H  toward  the  same  base  AB. 

In  Hke  manner  is  shown  that  the  aforesaid  obtuse 
angle  at  H  is  greater  than  the  obtuse  angle  at  the  point  D. 

And  thus  ever,  proceeding  toward  the  points  X. 

Finally  the  third  part  requires  a  longer  disquisition. 
If  therefore  it  can  be  done,  let  there  be  assigned  (fig.  34) 
a  certain  angle  MNC,  than  which  is  always  greater,  or 
anyhow  not  less,  the  excess  of  any  of  the  aforesaid  obtuse 
angles  above  a  right  angle.  It  follows  (from  P.  XXI.) 
that  the  sides  NM,  NC  comprehending  that  angle  MNC 
can  be  so  produced  that  the  perpendicular  MC  from  a 
certain  point  M  of  MN  [61]  let  fall  upon  NC  may  be 

»5» 


thesi  anguli  acuti)  qualibet  finita  assignata  longitudine, 
ut  puta  praedicta  basi  AB.  Hoc  stante:  assumatur  in 
BX  (fig.  35.)  quaedam  BT  aequalis  ipsi  CN;  educaturque 
ex  puncto  T  versus  AX  perpendicularis  TS,  quae  nempe 
(ex  Scholio  post  XXIV.  hujus)  occurret  ipsi  AX  in  quo- 
dam  puncto  S.  Deinde  ex  puncto  S  demittatur  ad  AB 
perpendicularis  SQ.  Cadet  haec  (propter  17.  primi)  ad 
partes  anguH  acuti  SAB  inter  puncta  A,  et  B.  Porro 
acutus  erit  angulus  QST  in  quadrilatero  QSTB,  cum  re- 
liqui  tres  anguli  sint  recti;  ne  (contra  V.  et  VI.  hujus) 
incidamus  in  hypothesin  aut  anguH  recti,  aut  anguh  ob- 
tusi.  Hinc  recta  SQ  major  erit  (ex  Cor.  I.  post  3.  hujus) 
recta  BT,  sive  CN;  ac  rursum  angulus  ASQ  major  erit 
excessu,  quo  angulus  obtusus  AST  excedit  angulum  rec- 
tum,  et  sic  major  angulo  MNC.  Ducatur  igitur  quaedam 
SF  secans  AQ  in  F,  et  efficiens  cum  SA  angulum  aequa- 
lem  ipsi  MNC.  Deinde  ex  puncto  A  ducatur  ad  SF  pro- 
ductam  perpendicularis  AO.  Cadet  punctum  O  (ex  17. 
primi)  infra  punctum  F,  cum  angukis  AFS  (ex  16.  ejus- 
dem  primi)  sit  obtusus.     Tandem  vero;  cum  FS  major 


152 


greater  (even  in  the  hypothesis  of  acute  angle)  than 
any  assigned  finite  length,  as  for  instance  the  aforesaid 
base  AB. 


Fig.  34, 


Fig.  35. 


This  standing;  assume  in  BX  (fig.  35)  a  certain  BT 
equal  to  CN,  and  erect  f  rom  the  point  T  toward  AX  the 
perpendicular  TS,  which  obviously  (from  Scholion  after 
P.  XXIV.)  meets  AX  in  a  certain  point  S.  Then  from 
the  point  S  let  fall  to  AB  the  perpendicular  SQ. 

This  falls  (because  of  Eu.  1. 17)  toward  the  parts  of 
the  acute  angle  SAB  between  the  points  A  and  B.  Again, 
acute  will  be  the  angle  QST  in  the  quadrilateral  QSTB, 
since  the  remaining  three  angles  are  right;  else  (against 
Propp.  V.  and  VI.)  we  come  upon  the  hypothesis  either 
of  right  angle  or  of  obtuse  angle. 

Hence  the  straight  SQ  will  be  greater  (from  Cor.  I. 
to  P.  III.)  than  the  straight  BT,  or  CN;  and  again  the 
angle  ASQ  will  be  greater  than  the  excess  by  which  the 
obtuse  angle  AST  exceeds  a  right  angle,  and  thus  greater 
than  the  angle  MNC.  Draw  therefore  a  certain  SF  cut- 
ting  AQ  in  F  and  making  with  SA  an  angle  equal  to 
MNC.  Then  from  the  point  A  draw  to  SF  produced 
the  perpendicular  AO.  The  point  O  falls  (from  Eu. 
I.  17)  below  the  point  F,  since  the  angle  AFS  (by  Eu. 
I.  16)  is  obtuse. 


1S3 


sit  (ex  19.  primi)  ipsa  QS,  et  sic  multo  major  ipsa  BT, 
sive  CN;  sumatur  in  FS  portio  IS  aequalis  ipsi  CN,  et 
ex  puncto  I  erigatur  ad  FS  perpendicularis  IR  occurrens 
in  puncto  R  ipsi  AS.  Cadet  autem  punctum  R  inter 
puncta  A,  et  S :  si  enim  caderet  in  aliquod  punctum  ipsius 
AF,  haberemus  in  eodem  triangulo  (contra  17.  primi) 
duos  angulos  majores  duobus  rectis,  cum  angulus  ad 
punctum  F  versus  partes  puncti  A  ostensus  jam  sit  ob- 
tusus. 

Post  tantum  apparatum  sic  concludo.  Quandoquidem 
in  quadrilatero  AOIR  recti  sunt  anguli  ad  puncta  O,  et  I ; 
et  est  acutus  angulus  (ex  17.  primi)  ad  punctum  A,  prop- 
ter  rectum  angulum  AOS;  ac  rursum  est  obtusus  (ex  16. 
[62]  ejusdem  primi)  angulus  IRA,  cum  rectus  sit  angulus 
RIS :  consequens  tandem  est  (ex  Cor.  11.  post  3.  hujus) 
ut  latus  AO  majus  sit  latere  IR.  At  (juncta  OQ)  latus 
AQ  majus  est  (ex  18.  primi)  latere  AO  propter  angulum 
obtusum  in  O,  cum  angulus  AOS  factus  sit  rectus.  Igi- 
tur  recta  AQ  multo  major  erit  recta  IR,  sive  (ex  26. 
primi)  recta  MC,  et  sic  multo  major  recta  AB,  pars  toto; 
quod  est  absurdum. 

Non  igitur  uUus  assignari  potest  angulus  MNC,  quo 
semper  major  sit,  aut  saltem  non  minor  excessus  cujusvis 
ex  praedictis  angulis  obtusis  supra  angulum  rectum. 
Quare  anguli  illi  obtusi,  magis  ac  magis  distantes  ab  eo 
puncto  A,  semper  magis  sine  ullo  certo  limite  accedent  ad 
aequalitatem  cum  angulo  recto.  Quod  erat  postremo  loco 
demonstrandum. 


IS4 


Finally,  however;  since  FS  is  greater  (by  Eu.  I.  19) 
than  QS  and  so  much  greater  than  BT  or  CN,  assume 
in  FS  the  piece  IS  equal  to  CN,  and  from  the  point  I 
erect  to  FS  the  perpendicular  IR  meeting  AS  in  the 
point  R. 

But  the  point  R  falls  between  the  points  A  and  S: 
for  if  it  fell  on  any  point  of  AF,  we  would  have  in  the 
same  triangle  (against  Eu.  I.  17)  two  angles  greater 
than  two  right  angles,  since  the  angle  at  the  point  F 
toward  the  parts  of  the  point  A  has  already  been  shown 
obtuse. 

After  so  much  preparation  thus  I  conclude.  Since 
in  the  quadrilateral  AOIR  the  angles  at  the  points  O  and  I 
are  right,  and  the  angle  at  the  point  A  (by  Eu.  I.  17) 
is  acute  because  of  the  right  angle  AOS,  and  again  the 
angle  IRA  (by  Eu.  I.  16)  is  obtuse,  [62]  since  the  angle 
RIS  is  right:  the  consequence  finally  is  (by  Cor.  II.  to 
P.  III.)  that  the  side  AO  is  greater  than  the  side  IR. 

But  (OQ  joined)  the  side  AQ  is  greater  (by  Eu. 
I.  19)  than  the  side  AO,  because  of  the  obtuse  angle  at 
O,  since  the  angle  AOS  was  made  right. 

Therefore  the  straight  AQ  will  be  much  greater  than 
the  straight  IR,  or  (by  Eu.  I.  26)  than  the  straight  MC, 
and  so  much  greater  than  the  straight  AB,  the  part  than 
the  whole;  which  is  absurd. 

Therefore  it  is  not  possible  to  assign  any  one  angle 
MNC,  than  which  always  is  greater,  or  anyhow  not  less, 
the  excess  of  each  of  the  aforesaid  obtuse  angles  above 
a  right  angle. 

Wherefore  those  obtuse  angles,  more  and  more  dis- 
tant  from  this  point  A,  ever  more  without  any  certain 
limit  approach  to  equality  with  a  right  angle. 

Quod  erat  postremo  loco  demonstrandum. 


I9f 


COROLLARIUM. 

Hoc  auteiii  stante,  quod  postremo  loco  demonstratum 
est,  manifeste  consequitur,  duas  illas  AX,  BX,  in  infini- 
tum  protractas,  commune  tandem  habituras,  vel  in  duobus 
distinctis  punctis,  vel  in  uno,  eodemque  puncto  X  infinite 
dissito,  perpendiculum.  Rursum  vero,  quod  non  in  duo- 
bus  distinctis  punctis  haberi  possit  commune  istud  per- 
pendiculum,  ex  eo  manifeste  liquet;  quia  caeterum  (ex 
Cor.  II.  post  XXIII.  hujus)  inciperent  inde  illae  rectae 
invicem  dissilire,  et  sic  neque  ad  infinitam  distantiam 
inter  se  concurrerent ;  quin  etiam  (contra  expressam  sup- 
positionem)  non  ad  se  invicem,  sine  ullo  certo  limite, 
semper  magis  versus  eas  partes  accederent.  Itaque  in 
uno,  eodemque  puncto  X  infinite  dissito  commune  habe- 
rent  perpendiculum.  [63] 


PROPOSITIO  XXIX. 

Resumpta  fig.  {2>Z.)  praecedentis  Propositionis :  Dico  om- 
nem  rectam  AC,  quae  secet  angulum  BAX,  ali- 
quando  ad  finitam,  seu  terminatam  distantiam  {etiam 
in  hypothesi  anguli  acuti)  occursuram  ipsi  BX  in 
quodam  puncto  P,  dum  nempe  illa  AC  semper  magis 
protrahatur  versus  partes  punctorum  X. 

Demonstratur.  Et  primo  quidem  (ne  recta  AC  spa- 
tium  claudat  cum  ea  AX)  occurret  ipsa  ad  finitam  distan- 
tiam  rectis  LK,  HK,  DK  in  quibusdam  punctis  C,  N,  M ; 
occurret,  inquam,  nisi  antea  (ad  finitam  utique  distan- 
tiam,  prout  intendimus)  occurrat  ipsi  BX  in  aliquo 
puncto  inter  punctum  B,  et  unum  aliquod  punctorum  K 
constituto.  Deinde  (ex  Cor.  I.  post  XXIII.  hujus)  ob- 
tusi  erunt  anguli  ACK,  ANK,  AMK.  Praeterea  anguli 
isti,  semper  obtusi,  accedent  (ex  praecedente)  sine  ullo 

156 


COROLLARY. 

But  this  standing,  which  in  the  last  case  was  demon- 
strated,  it  manifestly  follows  that  those  straights  AX, 
BX,  produced  infinitely  will  finally  have,  either  in  two 
distinct  points,  or  in  one  same  point  X  infinitely  distant, 
a  common  perpendicular. 

But  again,  that  this  common  perpendicular  cannot  be 
had  in  two  distinct  points  flows  manifestly  from  this,  be- 
cause  otherwise  (by  Cor.  II.  to  P.  XXIII.)  those  straights 
would  thence  begin  mutually  to  separate,  and  so  not  meet 
each  other  at  an  infinite  distance;  so  that  also  (against 
the  express  supposition)  they  would  not  mutually  ap- 
proach  each  other  without  any  certain  limit  ever  more 
toward  those  parts. 

So  they  must  have  the  common  perpendicular  in  one 
same  point  X  infinitely  distant.   [63] 

PROPOSITION  XXIX. 

Resuming  fig.  ?i2>  of  the  preceding  proposition:  I  say 
every  straight  AC,  which  cuts  angle  BAX,  finally 
at  a  finite,  or  terminated  distance  (even  in  the  hy- 
pothesis  of  acute  angle)  will  meet  BX  in  a  certain 
point  P,  if  only  AC  he  produced  ever  more  toward 
the  parts  of  the  points  X. 

Proof.  And  first  indeed  (lest  straight  AC  include 
space  with  AX)  it  must  meet  at  finite  distance  the 
straights  LK,  HK,  DK  in  certain  points  C,  N,  M ;  must 
meet,  I  say,  unless  before  (and  that  at  a  finite  distance, 
just  as  we  maintain)  it  meets  BX  in  some  point  between 
the  point  B  and  one  of  the  points  K. 

Then  (from  Cor.  I.  to  P.  XXIII.)  the  angles  ACK, 
ANK,  AMK  will  be  obtuse. 

Moreover    those    angles,    always    obtuse,    approach 

157 


certo  limite  ad  aequalitatem  cum  angulo  recto,  quoties 
nempe  illa  AC  non  nisi  ad  infinitam  distantiam  occursura 
putetur  ipsi  BX.  Igitur  deveniri  posset  ad  talem  ordina- 
tam  KMD,  ad  quam  angulus  AMK  minus  superaret  an- 
gulum  rectum,  quam  sit  ille  angulus  DAC.  Tunc  autem 
angulus  DAC,  sive  DAM,  una  cum  angulo  AMD  major 
erit  uno  recto.  Quare ;  addito  obtuso  angulo  ADM ;  tres 
simul  anguli  trianguli  ADM  majores  erunt  duobus  rectis, 
quod  est  contra  hypothesin  anguH  acuti.  Igitur  omnis 
recta  AC,  quae  secet  illum  angulum  BAX,  aliquando  ad 
finitam,  seu  terminatam  distantiam  (etiam  in  hypothesi 
anguli  acuti)  occurret  ipsi  BX  in  quodam  puncto  P. 
Quod  etc.  [64] 

COROLLARIUM  L 

Hinc  nulla  AZ,  quae  versus  partes  punctorum  X  angu- 
lum  acutum  efficiat  majorem  illo  BAX,  occurrere  unquam 
poterit,  sive  ad  finitam  sive  ad  infinitam  distantiam  ipsi 
BX.  Quatenus  enim  ita  contingeret,  jam  illa  AX,  divi- 
dens  angulum  BAZ,  deberet  (contra  praemissam  suppo- 
sitionem)  ad  finitam  distantiam  occurrere  ipsi  BX ;  prout 
demonstratum  id  est  de  recta  AC  dividente  angulum 
BAX. 

COROLLARIUM  II. 

Praeterea  sequitur  nullum  fore  determinatum  acutum 
angulum  omnium  maximum,  sub  quo  educta  ex  puncto 
A  ad  finitam  distantiam  occurrat  illi  BX.  Si  enim  ver- 
sus  partes  puncti  X  punctum  quodvis  assumas,  quod  sit 
altius  puncto  P,  constat  rectam  jungentem  punctum  A 


>9t 


(from  the  preceding  proposition)  without  any  certain 
limit,  to  equality  with  a  right  angle,  when  indeed  that 
AC  is  supposed  to  meet  BX  only  at  an 
infinite  distance. 

Therefore  such  an  ordinate  KMD 
can  be  reached  that  at  it  the  angle 
AMK  exceeds  a  right  angle  by  less 
than  the  angle  DAC.  But  then  angle 
DAC,  or  DAM,  together  with  angle 
AMD  will  be  greater  than  a  right  angle. 

Wherefore  the  obtuse  angle  ADM  A* 
being  added,  the  three  angles  together  ^^' 

of  the  triangle  ADM  will  be  greater  than  two  right 
angles,  which  is  against  the  hypothesis  of  acute  angle. 

Therefore  every  straight  AC,  which  cuts  that  angle 
BAX,  finally  at  a  finite  or  terminated  distance  (even  in 
the  hypothesis  of  acute  angle)  must  meet  BX  in  a  certain 
point  P. 

Quod  etc.  [64] 

COROLLARY  L 

Hence  no  straight  AZ,  which  toward  the  parts  of  the 
points  X  makes  an  acute  angle  greater  than  BAX  can 
ever  meet  BX,  either  at  a  finite  or  at  an  infinite  distance. 

For  as  far  as  so  should  happen,  now  AX,  dividing 
angle  BAZ,  ought  (against  the  premised  supposition) 
to  meet  BX  at  a  finite  distance,  as  this  is  demonstrated 
of  the  straight  AC  dividing  angle  BAX. 

COROLLARY  IL 

Moreover  it  follows  that  no  determinate  acute  angle 
will  be  the  maximum  of  all  under  which  a  straight  line 
produced  f  rom  point  A  meets  BX  at  finite  distance. 

For  if  toward  the  parts  of  the  point  X  you  assume 
any  point  higher  than  point  P,  it  follows  that  the  straight 

199 


cum  illo  puncto  altiore  majorem  angulum  effecturam  cum 
ipsa  AB,  quam  sit  angulus  BAP.  Atque  ita  semper  sine 
ullo  termino  intrinseco.  Quare  angulus  BAX  (dum  scili- 
cet  ipsa  AX,  et  semper  accedat  ad  eam  BX,  et  non  nisi 
ad  infinitam  distantiam  in  eandem  incidat)  erit  limes  ex- 
trinsecus  acutorum  omnium  angulorum,  sub  quibus  rectae 
eductae  ex  illo  puncto  A  ad  finitam  distantiam  occurrunt 
praedictae  BX. 

PROPOSITIO  XXX. 

Cuivis  terminatae  AB  insistat  ad  perpendiculum  (fig.  36.) 
quaedam  indefinita  BX.  Dico  primo  rectam  AY, 
perpendicidariter  elevatam  versus  partes  easdem  su- 
per  illa  AB,  fore  limitem  unum  intrinsecum  earum 
omnium,  quae  ex  illo  puncto  [65]  A  versus  easdem 
partes  eductae  commune  aliquod  {juxta  hypothesin 
anguli  acuti)  in  duobus  distinctis  punctis  obtinent 
perpendiculum  cum  altera  indefinita  BX.  Dico  se- 
cundo  nullum  fore  acutum  angulum  omnium  mini- 
mum,  sub  quo  educta  ex  praedicto  puncto  A  com- 
mune  aliquod  (juxta  praedictam  hypothesin)  in  duo- 
bus  distinctis  punctis  obtineat  perpendiculum  cum 
eadem  BX. 

Demonstratur  prima  pars.     Quoniam  enim  illa  AY 
commune  obtinet  cum  altera  BX  perpendiculum  AB  in 


i6o 


joining  point  A  with  this  higher  point  will  make  with 
AB  a  greater  angle  than  angle  BAP. 

And  so  ever  without  any  intrinsic  end. 

Wherefore  angle  BAX  (since  indeed  AX  both  always 
approaches  to  BX,  and  meets  it  only  at  an  infinite  dis- 
tance)  will  be  the  outside  Hmit  of  all  acute  angles  under 
which  straights  produced  from  that  point  A  meet  the 
aforesaid  BX  at  a  finite  distance. 


PROPOSITION  XXX. 

To  any  terminated  straight  AB  stands  at  right  angles 
(fig.  36)  a  certain  unbounded  straight  BX.  I  say 
firstly,  that  the  straight  A  Y,  erected  perpendicularly 
toward  the  same  parts  upon  AB,  will  be  one  intrinsic 
limit  of  all  those  straights,  which  drawn  from  the 


\- 

•i 

L 

i 

L 

j 

X 

K 

T 

/  /  / 

D  ^^-^^^^ 

\ 

H 

/"/ 

/H  ys 

\ 

i 

<^ 

/^ 

Fig.  2>(i. 

point  [65]  A  out  toward  the  same  parts  have  (in  the 
hypothesis  of  acute  angle)  a  common  perpendicular 
in  two  distinct  points  with  the  other  unbounded 
straight  BX.  I  say  secondly,  that  no  acute  angle 
will  be  the  minimum  of  all,  produced  under  which 
a  straight  from  the  aforesaid  point  A  (in  the  afore- 
said  hypothesis)  has  in  two  distinct  points  a  common 
perpendicular  with  BX. 

Proof  of  the  First  Part.     For  since  AY  has  in 
common  at  two  distinct  points  A  and  B  the  perpendicular 

i6i 


duobus  distinctis  punctis  A,  et  B ;  si  educatur  versus  eas- 
dem  partes  sub  angulo  obtuso  quaepiam  AZ,  constat  nul- 
lum  ad  eas  partes  esse  posse  in  duobus  distinctis  punctis 
commune  perpendiculum  ipsarum  AZ,  BX;  ne  scilicet 
ex  consecuturo  quadrilatero  continente  quatuor  angulos 
majores  quatuor  rectis  incidamus  (ex  XVI.  hujus)  in 
hypothesin  jam  reprobatam  anguli  obtusi,  contra  suppo- 
sitam  hoc  loco  hypothesin  anguH  acuti.  Igitur  illa  per- 
pendicularis  AY  erit  ex  ista  parte  limes  intrinsecus  earum 
omnium,  quae  ex  illo  puncto  A  versus  easdem  partes 
eductae  commune  aliquod  (juxta  illam  hypothesin  anguli 
acuti)  in  duobus  distinctis  punctis  obtineant  perpendicu- 
lum  cum  altera  indefinita  BX.     Quod  erat  primum. 

Demonstratur  secunda  pars.  Si  enim  fieri  potest ;  esto 
quidam  angulus  acutus  omnium  minimus,  sub  quo  educta 
AN  commune  habeat  cum  illa  BX  in  duobus  distinctis 
punctis  perpendiculum  ND.  Tum  assumpto  in  BX  altiore 
puncto  K,  ex  eo  educatur  ad  BX  perpendicularis  KL,  ad 
quam  ex  puncto  A  demittatur  (juxta  12.  primi)  perpen- 
dicularis  AL.  Jam  vero,  si  haec  AL  occurrat  in  quodam 
puncto  S  ipsi  ND,  constat  sane  angulum  BAL  minorem 
fore  eo  BAN,  qui  propterea  non  erit  omnium  minimus, 
sub  quo  educta  AN  commune  habeat  cum  illa  BX  in  duo- 
bus  distinctis  punctis  perpendiculum  ND.  [66]  Porro 
autem  ab  ea  perpendiculari  AL  secari  praedictam  ND  in 
quodam  ejus  intermedio  puncto  S  sic  demonstratur. 

Et  primo  quidem  non  posse  ab  ea  AL  secari  ipsam 
BK  in  quodam  puncto  M  constare  absolute  potest  ex  17. 
primi,  ne  scilicet  in  eodem  triangulo  MKL  duos  habea- 
mus  angulos  rectos  in  punctis  K,  et  L ;  praeterquam  quod 
in  hoc  ipso  haberemus  intentum  contra  illum  angulum 
BAN,  ne  scilicet  in  hac  tali  ratione  censeatur  omnium 


i$t 


AB  with  BX;  if  any  straight  AZ  is  drawn  toward  the 
same  parts  under  an  obtuse  angle,  it  follows  there  can 
be  toward  these  parts  in  two  distinct  points  no  common 
perpendicular  to  AZ,  BX.  Otherwise  from  the  resulting 
quadrilateral  containing  four  angles  greater  than  four 
right  angles,  we  hit  (from  P.  XVI.)  upon  the  already 
rejected  hypothesis  of  obtuse  angle,  against  the  hypothesis 
of  acute  angle  in  this  place  assumed. 

Therefore  that  perpendicular  AY  will  be  from  that 
side  an  intrinsic  limit  of  all  the  straights  which  drawn 
from  the  point  A  toward  the  same  parts  have  (in  the 
hypothesis  of  acute  angle)  at  two  distinct  points  a  com- 
mon  perpendicular  with  the  other  unbounded  straight  BX 

Quod  erat  primum. 

Proof  of  the  Second  Part,  For  if  it  were  possible, 
let  a  certain  acute  angle  be  the  least  of  all,  drawn  under 
which  AN  has  with  BX  in  two  distinct  points  the  common 
perpendicular  ND.  Then  in  BX  a  higher  point  K  being 
assumed,  from  this  erect  to  BX  the  perpendicular  KL. 
upon  which  from  the  point  A  let  fall  (by  Eu.  I.  12)  the 
perpendicular  AL. 

But  now,  if  this  AL  meets  ND  in  any  point  S,  it  cer- 
tainly  follows  that  angle  BAL  will  be  less  than  BAN. 
which  therefore  will  not  be  the  least  of  all  drawn  under 
which  AN  has  with  BX  in  two  distinct  points  a  common 
perpendicular  ND.  [66] 

But  furthermore  that  the  aforesaid  perpendicular  ND 
is  cut  by  this  perpendicular  AL  in  some  intermediate 
point  of  it  S  is  thus  demonstrated. 

And  first  indeed,  that  BK  cannot  be  cut  by  AL  in  any 
point  M  follows  absolutely  from  Eu.  L  17,  since  other- 
wise  in  the  same  triangle  MKL  we  would  have  two  right 
angles  at  the  points  K  and  L,  apart  f rom  the  fact  that  in 
this  case  we  would  have  our  assertion  about  that  angle 
BAN,  that  it  is  not  in  such  circumstances  the  least  of  alL 

163  «I 


minimus.  Rursum  vero  nequit  AL  esse  continuatio  ipsius 
AN ;  quia  caeterum  in  quadrilatero  NDKL  quatuor  habe- 
remus  angulos  rectos,  contra  hypothesim  anguli  acuti. 
Sed  neque  eam  DN  protractam  secare  potest  in  quovis 
ulteriore  puncto  H;  quia  angulus  AHN  (ex  16.  primi) 
foret  acutus,  propter  suppositum  rectum  angulum  exter- 
num  AND;  ac  propterea  anguhis  DHL  foret  obtusus,  et 
sic  in  quadrilatero  DHLK  quatuor  haberemus  angulos, 
qui  simul  sumpti  majores  forent  quatuor  rectis,  contra 
praedictam  hypothesin  anguh  acuti.  Igitur  constat  ab 
ea  AL  secari  debere  angulum  BAN,  qui  propterea  nequit 
dici  omnium  minimus,  sub  quo  educta  AN  commune 
habeat  cum  illa  BX  in  duobus  distinctis  punctis  perpen- 
diculum  ND.  Quod  erat  secundo  loco  demonstrandum 
Itaque  constat  etc. 

COROLLARIUM. 

Inde  autem  observare  hcet,  quod  sub  angulo  minore 
BAL  obtinetur  (in  hypothesi  anguh  acuti)  commune 
LK  perpendiculum,  remotius  quidem  ab  iha  basi  AB, 
prout  constat  ex  ipsa  constructione,  sed  rursum  minus 
altero  viciniore  communi  perpendiculo  ND,  quod  obtine- 
tur  sub  angulo  majore  BAN.  Ratio  hujus  posterioris 
est,  [67]  qtiia  in  quadrilatero  LKDS  angukis  ad  punctum 
S  acutus  est  in  praedicta  hypothesi,  cum  rehqui  tres  sup- 
ponantur  recti.  Quare  (ex  Cor.  I.  post  3.  hujus)  latus 
LK  minus  erit  contraposito  latere  SD,  et  sic  multo  minus 
latere  ND. 


164 


But  again  AL  cannot  be  the  continuation  of  AN ;  be- 
cause  otherwise  in  the  quadrilateral  NDKL  we  would 
have  four  right  angles,  against  the  hypothesis  of  acute 
angle. 

But  neither  can  it  cut  DN  produced  in  any  exterior 
point  H;  because  angle  AHN  (from  Eu.  L  16)  would 
be  acute,  on  account  of  the  external  angle  AND  supposed 
right ;  and  therefore  angle  DHL  would  be  obtuse,  and  so 
in  the  quadrilateral  DHLK  we  would  have  four  angles, 
which  taken  together  would  be  greater  than  four  right 
angles,  against  the  aforesaid  hypothesis  of  acute  angle. 

Therefore  it  follows  that  the  angle  BAN  must  be  cut 
by  this  AL,  and  therefore  cannot  be  declared  the  least  of 
all,  drawn  under  which  AN  has  with  BX  in  two  distinct 
points  a  common  perpendicular  ND. 

Quod  erat  secundo  loco  demonstrandum.  Itaque  con- 
stat  etc. 

COROLLARY. 

But  hence  is  permitted  to  observe,  that  under  a  lesser 
angleBALis  obtained  (inthe  hypothesis  of  acute  angle)  a 
common  perpendicular  LK,  more  remote  indeed  from  the 
base  AB,  as  follows  from  the  construction,  but  moreover 
less  than  the  other  nearer  common  perpendicular  ND, 
which  is  obtained  under  a  greater  angle  BAN. 

The  reason  of  this  latter  is  [67]  because  in  the  quadri- 
lateral  LKDS  the  angle  at  the  point  S  is  acute  in  the 
aforesaid  hypothesis,  since  the  three  remaining  angles  are 
supposed  right. 

Wherefore  (from  Cor.  I.  to  P.  HI.)  the  side  LK 
will  be  less  than  the  opposite  side  SD,  and  so  much  less 
than  the  side  ND. 


165 


PROPOSITIO  XXXI. 

Jam  dico  nullum  fore  praedictorum  in  duobus  disHnctis 
punctis  communium  perpendiculorum  limitem  deter- 
minatum^  quo  minus  sub  minore,  ac  minore  acuto 
angulOj  ad  illud  punctum  A  constituto,  deveniri  sem- 
per  possit  (juxta  hypothesin  anguli  acuti)  ad  tale 
commune  in  duobus  distinctis  punctis  perpendiculum, 
quod  sit  minus  qualibet  assignata  longitudine  R, 

Demonstratur.  Quatenus  enim  aliter  res  se  habeat; 
si  ex  puncto  K  (recole  fig.  30.)  in  quantalibet  a  puncto 
B  distantia  in  ea  BX  assignato,  educatur  perpendicularis 
KL,  ad  quam  ex  puncto  A  (juxta  12.  primi)  demissa  in- 
telligatur  perpendicularis  AL,  deberet  ipsa  KL  major 
esse  ea  longitudine  R.  Ratio  autem  est;  quia  assumpto  in 
eadem  BX  altiore  puncto  Q,  ex  quo  educatur  ad  ipsam 
BX  perpendicularis  QF,  ad  quam  (juxta  eandem  12. 
primi)  demittatur  perpendicularis  AF,  deberet  haec  rur- 
sum  saltem  non  esse  minor  ea  longitudine  R.  Erit  autem 
KL  (ex  Cor.  praeced.  Prop.)  major  ipsa  QF.  Igitur  ea 
KL  major  foret  praedicta  longitudine  R.  Atque  ita  sem- 
per  altius  procedendo. 

Jam  vero :  si  illa  quantacunque  KB  divisa  intelligatur 
(prout  in  XXV.  hujus)  in  portiones  KK,  aequales  illi 
longitudini  R,  educanturque  ex  ilHs  punctis  K  perpen- 
diculares,  quae  occurrant  ipsi  AX  in  punctis  H,  D,  M; 
non  erunt  anguH  ad  haec  puncta,  versus  partes  puncti  L, 
aut  recti,  aut  obtusi ;  ne  in  aHquo  quadrilatero,  ut  puta 


i66 


PROPOSITION  XXXI. 

Now  I  say  there  will  he,  of  the  aforesaid  common  perpen- 
diculars  in  two  distinct  points,  no  determinate  limit, 
such  that  under  a  smaller  and  smaller  acute  angle 
made  at  the  point  A,  it  would  not  always  be  possible 
to  attain  (in  the  hypothesis  of  acute  angle)  to  such  a 
common  perpendicular  in  two  distinct  points  as  is 
less  than  any  assignahle  length  R. 

Proof.  For  in  so  far  as  the  thing  were  otherwise; 
if  from  the  point  K  (resume  fig.  30)  in  BX  assigned  at 
any  however  great  distance  from 
the  point  B,  a  perpendicular  KL 
is  erected,  to  which  from  point  A 
(by  Eu.  I.  12)  the  perpendicular 
AL  is  supposed  let  fall,  KL  ought 
to  be  greater  than  the  length  R. 
The  reason  is ;  because  a  higher 
point  Q  being  assumed  in  this  BX, 
from  which  is  erected  to  BX  the 
perpendicular  QF,  to  which  (by 
the  same  Eu.  I.  12)  a  perpendic- 

ular  AF  is  let  fall,  this  again  must  anyhow  not  be  less 
than  the  length  R. 

But  KL  (from  Cor.  to  preceding  proposition)  will 
be  greater  than  QF.  Therefore  KL  would  be  greater 
than  the  aforesaid  length  R.  And  so  ever  proceeding 
higher. 

But  now,  if  this  however  great  KB  is  supposed  divided 
(as  in  P.  XXV.)  into  portions  KK,  equal  to  the  length 
R,  and  from  these  points  K  perpendiculars  are  erected, 
which  meet  AX  in  points  H,  D,  M;  the  angles  at  these 
points,  toward  the  parts  of  the  point  L,  will  neither  be 
right  nor  obtuse;  lest  in  some  quadrilateral,  as  suppose 

167 


KM-[^]LK  quatuor  simul  anguli  aequales  sint,  aut  ma- 
jores  quatuor  rectis,  contra  hypothesim  anguH  acuti,  juxta 
quam  procedimus.  Omnes  igitur  hujusmodi  anguH  acuti 
erunt  versus  partes  puncti  L;  ac  propterea  omnes  itidem 
ad  illa  puncta  obtusi  versus  partes  puncti  A.  Quare  (ex 
Cor.  I.  post  3.  hujus)  praedictarum  perpendicularium 
minima  quidem  erit  KL  remotior  a  basi  AB,  maxima 
KM  propinquior  eidem  basi ;  rehquarum  vero  propinquior 
remotiore  semper  major  erit.  Igitur  (ex  mea  praeced. 
24.  ejusque  Coroll.)  quatuor  simul  anguH  quadrilateri 
KHLK  remotioris  a  basi  AB  majores  erunt  quatuor  simul 
anguHs  reHquorum  omnium  quadrilaterorum  eidem  basi 
proximiorum.  Quare  (prout  XXV.  hujus)  destructa 
maneret  hypothesis  anguH  acuti. 

Itaq ;  constat  nuHum  f ore  praedictorum  in  duobus  dis- 
tinctis  punctis  communium  perpendiculorum  Hmitem  de- 
derminatum,  quo  minus  sub  minore,  ac  minore  acuto  an- 
gulo,  ad  iHud  punctum  A  constituto,  deveniri  semper  pos- 
sit  (juxta  hypothesin  anguH  acuti)  ad  tale  commune  in 
duobus  distinctis  punctis  perpendiculum,  quod  sit  minus 
quaHbet  assignata  longitudine  R.     Quod  erat  etc. 

PROPOSITIO  XXXII. 

Jam  dico  unum  aliquem  fore  (in  hypothesi  anguli  acuti) 
determinatum  acutum  angulum  BAX,  sub  quo  educta 
AX  (fig.  33.)  non  nisi  ad  infinitam  distantiam  inci- 
dat  in  eam  BX,  ac  propterea  sit  ipsa  limes  partim 
intrinsecus,  partim  extrinsecus.;  tum  earum  omnium, 
quae  sub  minoribus  acutis  angulis  ad  finitam  distan- 
tiam  incidunt  in  praedictam  BX;  tum  etiam  aliarum, 


i68 


KMLK,  [68]  the  four  angles  together  should  be  equal  to 
or  greater  than  four  rights,  contrary  to  the  hypothesis 
of  acute  angle,  according  to  which  we  are  proceeding. 
Therefore  all  such  angles  will  be  acute  toward  the  parts 
of  the  point  L ;  and  therefore  in  hke  manner  all  at  these 
points  obtuse  toward  the  parts  of  the  point  A.  Where- 
fore  (from  Cor.  I  to  P.  IIL)  of  the  aforesaid  perpen- 
diculars  the  least  will  indeed  be  KL  more  remote  from 
the  base  AB,  the  greatest  KM  nearer  this  base. 

And  of  the  remaining  the  nearer  will  be  ever  greater 
than  the  more  remote. 

Therefore  (from  the  preceding  P.  XXV,  and  its  corol- 
lary)  the  four  angles  together  of  the  quadrilateral  KHLK 
more  remote  from  base  AB  will  be  greater  than  the  four 
angles  together  of  all  the  remaining  quadrilaterals  nearer 
to  this  base.  Wherefore  (as  in  P.  XXV.)  the  hypothesis 
of  acute  angle  would  be  destroyed. 

Therefore  it  holds,  that  of  the  aforesaid  common  per- 
pendiculars  in  two  distinct  points  there  will  be  no  deter- 
minate  Hmit,  such  that  under  a  smaller  and  smaller 
acute  angle  made  at  the  point  A,  it  would  not  always  be 
possibleto  attain  (inthe  hypothesis  of  acute  angle)  to  such 
a  common  perpendicular  in  two  distinct  points  as  may  be 
less  than  any  assigned  length  R. 

Ouod  erat  demonstrandum. 

PROPOSITION  XXXII. 

Now  I  say  there  is  (in  the  hypothesis  of  actite  angle)  a 
certain  determinate  acute  angle  BAX  drazvn  under 
which  AX  (fig.  33)  only  at  an  infinite  distance  meets 
BX,  and  thus  is  a  limit  in  part  from  within,  in  part 
from  without;  on  the  one  hand  of  all  those  which 
under  lesser  acute  angles  meet  the  aforesaid  BX  at 
a  finite  distance;  on  the  other  hand  also  of  the  others 


169 


quae  sub  majorihus  angulis  acutis,  usque  ad  angulum 
rectum  inclusive,  commune  obtinent  in  duobus  dis- 
tinctis  punctis  perpendiculum  cum  eadem  BX.  [69] 

Demonstratur.  Nam  primo  constat  (ex  Cor.  II.  post 
XXIX.  hujus)  nullum  fore  determinatum  acutum  angu- 
lum,  omnium  maximum,  sub  quo  educta  ex  illo  puncto 
A  ad  finitam  distantiam  occurrat  praedictae  BX.  Secundo 
constat  nullum  itidem  esse  (in  hypothesi  anguli  acuti) 
acutum  angulum  omnium  minimum,  sub  quo  educta  coni- 
mune  habeat  in  duobus  distinctis  punctis  perpendiculum 
cum  illa  BX;  quandoquidem  (ex  praecedente)  nullus  esse 
potest  Hmes  determinatus,  quo  minus  sub  minore  acuto 
angulo  ad  illud  punctum  A  constituto  deveniri  possit  ad 
tale  commune  in  duobus  distinctis  punctis  perpendiculum, 
quod  sit  minus  quahbet  assignabiH  longitudine  -R. 

Atque  hinc  tertio  consequitur  unum  ahquem  (in  ea 
hypothesi)  esse  debere  determinatum  acutum  angulum 
BAX,  sub  quo  educta  AX  ita  semper  magis  accedat  ad 
eam  BX,  ut  non  nisi  ad  infinitam  distantiam  in  eandem 
incidat. 

Porro  autem  hanc  ipsam  AX  fore  Hmitem  partim  in- 
trinsecum,  partim  extrinsecum  utriusque  praedictarum 
rectarum  classis,  sic  demonstratur.  Nam  primo  conveniet 
cum  iUis  rectis,  quae  ad  finitam  distantiam  occurrunt  ipsi 
BX,  cum  ipsa  etiam  aHquando  conveniat;  discrepabit  au- 
tem,  quia  ipsa  non  nisi  ad  infinitam  distantiam.  Secundo 
autem  conveniet  etiam,  et  simul  discrepabit  ab  iUis  rectis, 
quae  commune  obtinent  in  duobus  distinctis  punctis  per- 
pendiculum  cum  illa  BX ;  quia  ipsa  etiam  commune  obti- 
net  perpendiculum  cum  eadem  BX ;  sed  in  uno  eodemque 
puncto  X  infinite  dissito.    Hoc  autem  postremum  censeri 


which  under  greater  acute  angles,  even  to  a  right 
angle  inclusive,  have  a  common  perpendicular  in  two 
distinct  points  with  BX.  [69] 

Proof.  First  it  holds  (from  Cor.  II.  to  P.  XXIX.) 
that  no  determinate  acute  angle  will 
be  the  greatest  of  all  drawn  under 
which  a  straight  from  the  point  A 
meets  the  aforesaid  BX  at  a  finite 
distance. 

Secondly,  it  holds  in  like  manner 
that  (inthehypothesis  of  acute  angle) 
no  acute  angle  will  be  the  least  of  all 
drawn  under  which  a  straight  has  a  a^ 
common  perpendicular  in  two  distinct  ^^* 

points  with  BX;  since  indeed  (from  what  precedes) 
there  can  be  no  determinate  Hmit,  such  that  there  cannot 
be  found,  under  a  lesser  angle  constituted  at  the  point  A, 
a  common  perpendicular  in  two  distinct  points,  which  is 
less  than  any  assignable  length  R. 

And  hence  follows  thirdly,  that  (in  this  hypothesis) 
there  must  be  a  certain  determinate  acute  angle  BAX, 
drawn  under  which  AX  so  approaches  ever  more  to  BX, 
that  only  at  an  infinite  distance  does  it  meet  it. 

But  further  that  this  AX  is  a  Hmit  in  part  f rom  within 
in  part  from  without  of  each  of  the  aforesaid  classes  of 
straights  is  proved  thus.  First,  it  agrees  with  those 
straights  which  meet  BX  at  a  finite  distance  since  it  also 
finally  meets;  but  it  differs,  because  it  meets  only  at  an 
infinite  distance. 

But  secondly  it  also  agrees  with,  and  at  the  same  time 
diflFers  from  those  straights  which  have  a  common  per- 
pendicular  in  two  distinct  points  with  BX ;  because  it  also 
has  a  common  perpendicular  with  BX ;  but  in  one  and  the 
same  point  X  infinitely  distant.     But  this  latter  ought  to 

171 


debet  demonstratum  in  XXVIII.  hujus,  prout  moneo  in 
ejusdem  Corollario. 

Itaque  constat  unum  aliquem  fore  (in  hypothesi  an- 
guli  acuti)  determinatum  acutum  angulum  BAX,  sub 
quo  educta  AX  non  nisi  ad  infinitam  distantiam  incidat  in 
[70J  eam  BX,  ac  propterea  sit  ipsa  limes  partim  intrin- 
secus,  partim  extrinsecus;  tum  earum  omnium,  quae  sub 
minoribus  acutis  anguHs  ad  finitam  distantiam  incidunt  in 
praedictam  BX;  tum  etiam  aliarum,  quae  sub  majoribus 
angulis  acutis,  usque  ad  angulum  rectum  inclusive,  com- 
mune  obtinent  in  duobus  distinctis  punctis  perpendiculum 
cum  eadem  BX.     Quod  erat  etc. 

PROPOSITIO  XXXIII. 

Hypothesis  anguli  acuti  est  ahsolute  falsa;  quia  repugnans 
naturae  lineae  rectae. 

Demonstratur,  Ex  praemissis  Theorematis  constare 
potest  eo  tandem  perducere  Geometriae  Euchdeae  inimi- 
cam  hypothesin  anguh  acuti,  ut  agnoscere  debeamus  duas 
in  eodem  plano  existentes  rectas  AX,  BX,  quae  in  infini- 
tum  protractae  versus  eas  partes  punctorum  X  in  unam 
tandem  eandemque  rectam  hneam  coire  debeant,  nimi- 
rum  recipiendo,  in  uno  eodemque  infinite  dissito  puncto 
X,  commune  in  eodem  cum  ipsis  plano  perpendicukim. 
Quoniam  vero  de  primis  ipsis  principiis  agendum  mihi 
hic  est,  dihgenter  curabo,  ut  nihil  omittam  quasi  nimis 
scrupulose  objectum,  quod  quidem  exactissimae  demon- 
strationi  opportunum  esse  cognoscam. 

LEMMA  I. 

Duae  rectae  lineae  spatium  non  comprehendunt. 

Definit  Euchdes  hneam  rectam,  quae  ex  aequo  sua 
interjacet  puncta.     Esto  igitur  (fig.  37.)  hnea  quaedam 

172 


be  considered  demonstrated  in  P.  XXVIII. ,  as  I  point 
out  in  its  corollary. 

Therefore  it  holds,  that  (in  the  hypothesis  of  acute 
angle)  there  will  be  a  certain  determinate  acute  angle 
BAX,  drawn  under  which  AX  only  at  an  infinite  distance 
meets  [70]  BX,  and  thus  is  a  hmit  in  part  from  within, 
in  part  f  rom  without ;  on  the  one  hand  of  ah  those  which 
under  lesser  acute  angles  meet  the  aforesaid  BX  at  a 
finite  distance ;  on  the  other  hand  also  of  the  others  which 
under  greater  acute  angles,  even  to  a  right  angle  inclusive, 
have  a  common  perpendicular  in  two  distinct  points 
with  BX. 

Quod  erat  etc. 

PROPOSITION  XXXIII. 
The  hypothesis  of  acute  angle  is  absolutely  false;  because 
repugnant  to  the  nature  of  the  straight  line. 

Proof.  From  the  foregoing  theorem  may  be  estab- 
Hshed,  that  at  length  the  hypothesis  of  acute  angle  inim- 
ical  to  the  EucHdean  geometry  has  as  outcome  that  we 
must  recognize  two  straights  AX,  BX,  existing  in  the 
same  plane,  which  produced  in  infinitum  toward  the  parts 
of  the  points  X  must  run  together  at  length  into  one  and 
the  same  straight  Hne,  truly  receiving,  at  one  and  the 
same  infinitely  distant  point  a  common  perpendicular  in 
the  same  plane  with  them. 

But  since  I  am  here  to  go  into  the  very  first  principles, 
I  shaU  diHgently  take  care,  that  I  omit  nothing  objected 
almost  too  scrupulously,  which  indeed  I  recognize  to  be 
opportune  to  the  most  exact  demonstration. 

LEMMA  I. 
Two  straight  lines  do  not  inclose  a  space. 

EucHd  defines  a  straight  Hne  as  one  which  lies  eve'"^'' 
between  its  points. 


AX,  quae  ex  puncto  A  per  sua  quaelibet  intermedia 
puncta  continuative  excurrat  usque  ad  punctum  X.  Non 
di-[71]cetur  haec  linea  recta,  si  talis  ipsa  fuerit,  ut  circa 
duo  illa  immota  extrema  sua  puncta  possit  ipsa  in  alteram 
partem  converti,  ut  puta  a  laeva  parte  in  dexteram :  Non 
dicetur,  inquam,  linea  recta;  quia  non  jacebit  ex  aequo 
inter  sua  designata  extrema  puncta ;  quandoquidem  vel  in 
laevam  partem  declinabit,  ubi  ex  puncto  A  excurrit  ad 
punctum  X  per  quaedam  intermedia  puncta  B;  vel  decli- 
nabit  in  dexteram,  ubi  ex  eodem  immoto  puncto  A  ex- 
currit  ad  idem  immotum  punctum  X  per  quaedam  inter- 
media  puncta  C,  quae  alia  plane  sunt  a  praedictis  punctis 
B.  Scilicet  illa  sola  linea  AX  dici  poterit  recta,  quae  ex- 
currat  ex  puncto  A  ad  punctum  X  per  talia  intermedia 
puncta  D,  quae  ipsa,  prout  sic  invicem  continuata,  revolvi 
nequeant,  circa  illa  immota  extrema  puncta  A,  et  X,  ad 
novum  et  novum  occupandum  situm. 

In  hac  autem  rectae  lineae  idea  manifeste  continetur 
proposita  veritas,  duas  nempe  rectas  lineas  spatium  non 
comprehendere.  Si  enim  duae  exhibeantur  lineae  clau- 
dentes  spatium,  quarum  nempe  communia  sint  extrema 
duo  puncta  A,  et  X,  facile  ostenditur  vel  neutram,  vel 
unam  tantum  illarum  linearum  esse  rectam.  Neutra  erit 
recta,  ut  puta  ABBX,  et  ACCX,  si  circa  duo  extrema  im- 
mota  puncta  A,  et  X,  ita  revolvi  posse  intelligantur  ipsae 
ABBX,  ACCX,  ut  reliqua  ipsarum  intermedia  puncta  ad 
novum,  et  novum  occupandum  locum  pertranseant.  Una 
tantum  erit  recta,  ut  puta  ADDX,  si  circa  illa  immota  ex- 
trema  puncta  ita  revolvi  intelligantur  ipsae  ABBX, 
ACCX,  quae  hinc  inde  cum  illa  ADDX  spatium  claudunt. 
ut  ipsarum  quidem  ABBX,  ACCX  puncta  intermedia  ad 


«74 


Let  there  be  therefore  (fig.  2>7)  any  line  AX,  which 
from  the  point  A  through  any  intermediate  points  of  it 
runs  consecutively  even  to  the  point  X.  This  Hne  is  not 
[71]  called  straight,  if  it  be  such,  that  it  can  be  turned 
about  its  two  end  points  into  another  region,  as  suppose 
f rom  the  left  region  into  the  right :  I  say  it  is  not  called 
a  straight  Hne;  because  it  will  not  He  ex  aeqiw  between 
its  designated  extreme  points;  since  either  it  wiH  lean 
toward  the  left  side,  where  from  the  point  A 
it  runs  out  to  the  point  X  through  certain  inter- 
mediate  points  B;  or  it  bends  to  the  right, 
where  from  the  same  fixed  point  A  it  runs  out 
to  the  same  fixed  point  X  through  certain 
intermediate  points  C  which  are  wholly  dif- 
ferent  f rom  the  aforesaid  points  B.  Obviously 
only  that  line  AX  can  be  called  straight,  which 
runs  out  from  the  point  A  to  the  point  X 
through  such  intermediate  points  D,  as,  in  order  one 
after  another  continued,  cannot  be  revolved,  about  those 
fixed  extreme  points  A,  and  X,  to  occupying  new  and 
new  location. 

But  in  this  idea  of  the  straight  line  is  contained  mani- 
festly  the  announced  truth,  namely  that  two  straight  lines 
do  not  inclose  a  space.  For  if  two  lines  are  shown  in- 
closing  a  space,  which  have  in  common  the  two  extreme 
points  A,  and  X,  it  is  easily  shown  either  that  neither, 
or  only  one  of  them  is  straight.  Neither  will  be  straight, 
as  for  example  ABBX,  and  ACCX,  if  it  be  supposed  so 
that  they  can  be  revolved  about  two  fixed  extreme  points 
A,  and  X,  that  their  remaining  intermediate  points  pass 
over  to  occupying  new  and  new  place. 

One  only  will  be  straight,  as  for  example  ADDX,  if 
about  those  fixed  end  points  we  may  suppose  ABBX, 
ACCX,  which  on  both  sides  with  that  ADDX  inclose  a 
space,  so  to  be  revolved,  that  indeed  the  intermediate  points 

^7S 


novum,  et  novum  occupandum  locum  pertranseant,  ipsius 
vero  ADDX  puncta  omnia  etiam  intermedia  in  eodem 
loco  persistant.  Non  igitvir  fieri  potest,  ut  duae  juxta 
praemissam  intelligentiam  rectae  lineae,  spatium  compre- 
hendant.     Quod  erat  propositum.  [72] 

COROLLARIUM  L 

Hinc  porro  sequitur  admitti  oportere  postulatum  illud 
Euclidaeum :  quod  a  dato  puncto  ad  qiwdlihet  assignatum 
punctum  rectam  lineam  ducere  liceat.  Nam  clare  intelli- 
gitur,  duas  semper  sine  ullo  certo  limite  duci  posse  lineas, 
praedictis  punctis  A,  et  X  terminatas,  quae  propiores  in- 
vicem  fiant,  minusque  idcirco  spatium  comprehendant, 
dum  sciHcet  una  quidem  ducatur  ad  laevam  partem,  et  al- 
tera  uniformis  ad  dexteram,  sive  una  sursum,  et  altera 
deorsum;  duci,  inquam,  posse  Hneas  ejusmodi  semper  in- 
vicem  sine  uUo  certo  Hmite  propiores,  quae  utique  omnino 
uniformes  inter  se  sint,  sibique  invicem  idcirco  succedant, 
dum  circa  immota  extrema  puncta  A,  et  X,  revolvi  ipsae 
inteHigantur.  Inde  autem  clare  itidem  inteUigitur,  sequi 
tandem  debere  (in  semper  majore  harum  uniformium  H- 
nearum,  unius  ad  alteram  accessu)  coitionem  in  unam,. 
eandemque  Hneam  ADX,  quae  circa  immota  extrema  iha 
puncta  revolvi  nequeat  ad  occupandum  novum  locum.  Et 
haec  erit  Hnea  recta  postulata. 

Ubi  rursum  constat  unicam  esse,  quae  a  dato  puncto 
ad  quodHbet  alterum  assignatum  punctum  potest  duci 
Hnea  recta. 


176 


of  ABBX,  ACCX,  pass  over  to  the  occupying  of  new  and 
new  position,  but  on  the  contrary  all  the  intermediate 
points  of  ADDX  remain  in  the  same  place. 

Therefore  it  cannot  be,  that  two  lines  straight  in 
accordance  with  the  premised  conception  inclose  a  space. 

Quod  erat  propositum.  [72] 

COROLLARY  I. 

Hence  moreover  follows  we  should  admit  the  Euclid- 
ean  postulate:  that  from  a  given  point  to  any  assigned 
point  a  straight  line  may  be  draivn. 

For  it  is  clearly  understood,  that  always  two  hnes 
without  any  certain  hmit  can  be  drawn,  terminated  in  the 
aforesaid  points  A,  and  X,  which  mutually  approach,  and 
therefore  inclose  less  space,  while  indeed  one  is  drawn 
toward  the  left  side,  and  the  other  of  the  same  shape 
toward  the  right,  or  one  over,  and  the  other  under;  I 
say,  lines  of  this  sort  may  be  drawn  always  mutually 
approaching  without  any  certain  Hmit,  which  are  com- 
pletely  of  the  same  shape  with  each  other,  and  therefore 
mutually  succeed  each  other  when  supposed  revolved 
about  the  fixed  end  points  A,  and  X. 

Whence  clearly  in  Hke  manner  is  understood,  at 
length  (in  ever  greater  approach  of  these  Hke  shaped 
Hnes,  one  to  the  other)  should  foHow  the  coalescence 
into  one,  and  the  same  Hne  ADX,  which  cannot  be  re- 
volved  about  those  fixed  extreme  points  so  as  to  occupy 
a  new  position.  And  this  wiH  be  the  straight  Hne  postu- 
lated. 

Where  again  is  estabHshed  to  be  unique  the  straight 
Hne,  which  can  be  drawn  f rom  a  given  point  to  any  other 
assigned  point. 


lyy 


COROLLARIUM  IL 

Praeterea  sequitur  uniformem  esse  debere  intelligen- 
tiam  alterius  Euclideae  definitionis,  in  qua  dicit  planam 
superficiem  esse,  quae  ex  aequo  suas  interjacet  lineas.  Si 
enim  superficies  clausa  praedictis  lineis  una  ADX  recta, 
et  altera  ABBX  (sive  haec  sit  unica,  aut  multiplex  linea 
curva,  sive  sit  composita  ex  duabus,  aut  pluribus  lineis 
rectis,  ut  puta  AB,  BB,  BX)  si,  inquam,  superficies  [73] 
ejusmodi  revolvi  intelligatur  circa  immotam  rectam  ADX, 
usque  dum  ipsa  linea  ABX  perveniat  ad  congruendum 
lineae  ACX,  in  parte  adversa  locatae,  quae  utique  ad  om- 
nimodam  aequalitatem,  et  similis  omnino  sit  ipsi  ABX, 
et  rursum  cum  eadem  recta  ADX  claudat  (versus  eandem 
sive  supernam,  sive  infernam  partem)  superficiem  om- 
nino  aequalem,  et  similem  antedictae;  alterutrum  sane 
continget;  vel  ita  ut  una  superficies  alteri  adamussim 
congruat;  vel  ita  ut  intra  duas  illas  superficies  claudatur 
spatium  trinae  dimensionis.  Et  primum  quidem  si  con- 
tingat,  dicetur  superficies  plana ;  sin  vero  contingat  secun- 
dum,  non  dicetur  superficies  plana;  quia  tunc  aliae  inter- 
mediae  intelligi  poterunt  inter  easdem  extremas  lineas 
interpositae  superficies  invicem  aequales,  ac  similes,  quae 
semper  magis  ad  se  invicem  sine  ullo  certo  limite  acce- 
dant,  ac  propterea  usque  ad  excludendum  omne  spatium 
intermedium.  Tunc  autem  utraque  illa  superficies  dicetur 
plana,  quia  vere  jacebit  ex  aequo  inter  suas  extremas 
lineas,  sine  ullo  ascensu,  aut  descensu  in  partes  adversas. 

LEMMA  II. 

Duae  lineae  rectae  non  possunt  habere  unum  et  idem 
segmentum  commune. 

Demonstratur.     Si  enim  fieri  potest;  unum  et  idem 
segmentum  AX  commune  sit   (fig.  38.)   duabus  rectis, 

178 


COROLLARY  IL 

Moreover  it  follows  the  interpretation  should  be  the 
same  of  the  other  EucHdean  definition,  in  which  he  says 
a  surface  is  plane,  which  lies  evenly  hetween  its  lines. 

For  if  a  surface  inclosed  by  the  aforesaid  lines  one 
ADX  straight,  and  another  ABBX  (whether  this  be  a 
simple  or  multiplex  curved  Hne,  or  be  composed  of  two, 
or  several  straight  Hnes,  as  suppose  AB,  BB,  BX)  if, 
I  say,  a  surface  [73]  of  this  sort  is  supposed  to  be  revolved 
about  the  fixed  straight  ADX,  until  the  Hne  ABX  comes 
to  congruence  with  the  Hne  ACX,  located  in  the  opposite 
part,  which  assuredly  is  in  every  way  equal  and  whoHy 
similar  to  ABX,  and  again  with  the  same  straight  ADX 
incloses  (toward  the  same  part,  whether  upper  or  under) 
a  surface  whoHy  equal,  and  similar  to  the  aforesaid :  one 
of  two  things  certainly  happens;  either  one  surface  fits 
the  other  completely;  or  between  those  two  surfaces  is 
inclosed  a  three-dimensional  space. 

And  indeed  if  the  first  happens,  the  surface  is  caHed 
plane;  but  if  the  second  happens  the  surface  is  not  caHed 
plane;  because  then  may  be  supposed  other  intermediate 
surfaces,  mutuaHy  equal,  and  similar,  interposed  between 
the  same  extreme  Hnes,  which  always  mutuaHy  approach 
more  to  each  other  without  any  certain  Hmit,  and  there- 
fore  even  to  the  exclusion  of  every  intermediate  space. 

But  then  each  surface  is  caHed  plane,  because  truly 
it  Hes  ex  aequo  between  its  extreme  Hnes,  without  any 
ascent  or  descent  into  bordering  parts. 

LEMMA  IL 

Two  straight  lines  cannot  have  one  and  the  same  segment 

in  common. 

Proof.   For  if  that  is  possible,  let  one  and  the  same 

segment  AX  be  common  (fig.  38)  to  the  two  straights 

179 


per  punctum  X  in  eodem  plano  continuatis  AXB,  et 
AXC.  Tum  centro  X,  et  intervallo  XB,  sive  XC,  descri- 
batur  arcus  BMC,  ad  cujus  quodlibet  punctum  M  junga- 
tur  ex  puncto  X  recta  XM. 

Dico  primo,  lineam  AXM  fore  et  ipsam,  in  facta 
hy-[74]pothesi,  Hneam  rectam,  ex  puncto  A  per  punctum 
X  continuatam.  Si  enim  Hnea  ejusmodi  recta  non  sit, 
duci  poterit  (ex  Cor.  I.  praecedentis  Lemmatis)  aHa  quae- 
dam  linea  AM,  quae  ipsa  sit  recta.  Haec  autem  vel  secabit 
in  aHquo  puncto  K  alterutram  ipsarum  XB,  XC;  vel 
earundem  alterutram,  ut  puta  eam  XB  claudet  intra  spa- 
tium  comprehensum  ipsis  AX,  XM,  et  APLM.  At  horum 
prius  manifeste  repugnat  praecedenti  Lemmati;  quia  sic 
duae  suppositae  rectae  Hneae,  una  AXK,  et  altera  ATK, 
spatium  clauderent.  Posterius  autem  uniformis  absurdi 
statim  convincitur. 

Nam  constat  rectam  XB,  si  per  B  ulterius  protrahatur, 
occursuram  tandem  in  aHquo  puncto  L  ipsi  APLM ;  unde 
rursum  duae  suppositae  rectae,  una  AXBL,  et  altera  APL, 
spatium  claudent.  Porro  uniforme  sequitur  absurdum, 
si  fingamus,  quod  recta  XB,  ulterius  protracta  per  B, 
occurrat  tandem  in  quovis  aHo  puncto  aut  rectae  XM,  aut 
rectae  XA. 

Ex  istis  autem  evidenter  consequitur  Hneam  AXM 
fore  ipsam,  in  facta  hypothesi,  Hneam  rectam  ex  puncto 
A  ad  punctum  M  deductam.    Quod  erat  propositum. 

Dico  secundo,  eam  suppositam  rectam  AXB  (quate- 
nus  quidem  inteHigatur  conservare  suam  iham  qualem- 


i8« 


Fig.  38. 


AXB,  and  AXC  produced  through  the  point  X  in  the 
same  plane.  Then  with  center  X,  and  radius  XB,  or  XC, 
describe  the  arc  BMC,  to  any  point  of  which  M  is  drawn 
from  the  first  point  X  the  straight  XM. 

I  say  first,  under  the  assumed  hypothesis  also  the  line 
AXM  will  be   U4]   a  straight  line, 
continued  f  rom  the  point  A  through 
the  point  X. 

For  if  a  line  of  this  sort  be  not 
straight,  there  can  be  drawn  (from 
Cor.  I.  of  the  preceding  lemma)  a 
certain  other  line  AM,  which  itself 
is  straight.  But  this  either  cuts  in 
some  point  K  one  or  the  other  of 
those  straights  XB,  XC;  or  it  in- 
closes  one  or  the  other  of  them,  as  suppose  XB  within 
the  space  bounded  by  AX,  XM,  and  APLM. 

But  the  first  of  these  is  manifestly  contrary  to  the 
preceding  lemma;  because  thus  two  lines  supposed 
straight,  one  AXK,  and  the  other  ATK,  would  inclose 
a  space. 

But  the  second  is  at  once  convicted  of  a  like  absurdity. 

For  it  is  certain  that  the  straight  XB,  if  produced  on 
through  B,  will  at  length  meet  this  APLM  in  some  point 
L ;  whence  again  two  lines  supposed  straight,  one  AXBL, 
and  the  other  APL,  will  inclose  a  space.  But  a  like  absurdity 
follows,  if  we  assume,  that  the  straight  XB,  produced  on 
through  B,  at  length  meets  in  some  other  point  either  the 
straight  XM,  or  the  straight  XA. 

But  from  this  evidently  follows  that  the  line  AXM 
is  itself,  in  the  assumed  hypothesis,  the  straight  line 
drawn  from  the  point  A  to  the  point  M. 

Quod  erat  propositum. 

I  say  secondly,  that  the  assumed  straight  AXB  (inas- 
much  as  it  is  understood  to  retain  its  arbitrary  continuation 


cunque  continuationem  ex  puncto  A  per  X  versus  B)  non 
posse  recipere  duplicem  aliam  in  eodem  plano  positionem, 
in  quarum  utraque  portio  quidem  AX  in  eodem  situ  per- 
sistat,  portio  vero  altera  XB  in  una  illarum  duarum  posi- 
tionum  congruat  (exempli  causa)  ipsi  XC,  et  in  alia  posi- 
tione  congruat  ipsi  XM. 

Scilicet  non  hic  renuo,  quin  portio  XB,  si  intelligatur 
moveri  in  illo  suo  plano  circa  punctum  X,  adeo  ut  succes- 
sive  adamussim  congruat  (ex  praecedente  Lemmate)  non 
modo  ipsis  XM,  XC,  verum  etiam  adamussim  con-[75] 
gruat  infinitis  aliis  rectis,  quae  ex  puncto  X  duci  possunt 
ad  reliqua  intermedia  puncta  arcus  BC :  Non,  inquam,  hic 
renuo,  quin  illa  XB  in  quaHbet  illarum  positionum  con- 
siderari  debeat  tanquam  continuatio  in  rectum  ipsius  im- 
motae  AX ;  cum  magis  circa  eam  AXM  jam  demonstrave- 
rim  id  secuturum  in  facta  hypothesi  ilHus  communis  seg- 
menti :  Unice  igitur  hic  assero,  in  una  tantum  novarum 
illarum  positionum,  ut  puta  dum  congruit  ipsi  XC,  reti- 
neri  ab  ea  posse  illam  eandem  qualemcunque  continuatio- 
nem,  quam  obtinet  in  prima  positione,  ubi  ex  puncto  A 
per  X  procedit  versus  punctum  B. 

Et  istud  quidem  sic  demonstratur.  Nam  primo  con- 
stat  continuationem  illam  AXB  nequire  esse  omnino  simi- 
lem,  aut  aequalem  continuationi  AXC,  si  utraque  con- 
sideretur  versus  eandem  seu  laevam,  seu  dexteram  par- 
tem;  quia  caeterum  in  ea  tali  positione  deberent  invicem 
congruere  ipsae  AXB,  AXC ;  quod  est  contra  hypothesim 
communis  illius  segmenti  AX :  Deberent,  inquam,  con- 
gruere;  dum  scilicet,  relate  ad  eam  immotam  AX,  aeque 
similiter  in  eandem  seu  laevam,  seu  dexteram  partem  con- 
vergerent  in  eo  tali  plano  illae  continuatae  XB,  et  XC. 
Secundo  constat  nihil  vetare,  quin  praedicta  continuatio 
AXB,  considerata  versus  unam  partem,  ut  puta,  ad  lae- 
vam,  similis  plane  sit,  aut  aequalis  continuationi  AXC, 
consideratae  versus  partem  adversam,  ut  puta,  ad  dexte- 

182 


from  the  point  A  through  X  toward  B)  cannot  have  two 
different  positions  in  the  same  plane,  in  both  of  which 
the  portion  indeed  AX  persists  in  the  same  place,  but  the 
other  portion  XB  in  one  of  those  two  positions  fits  (for 
example)  XC,  and  in  the  other  position  fits  XM. 

Of  course  I  do  not  here  deny,  that  the  portion  XB,  if 
it  is  supposed  to  be  moved  in  its  plane  about  the  point  X, 
so  that  successively  it  fits  exactly  (from  the  preceding 
lemma)  not  merely  XM,  XC,  but  also  exactly  fits  [75] 
the  other  infinitely  many  straights,  which  from  the  point 
X  may  be  drawn  to  the  remaining  intermediate  points  of 
the  arc  BC :  I  say,  I  do  not  here  deny,  that  XB  in  any  of 
its  positions  may  be  considered  as  the  continuation  in  a 
straight  of  that  fixed  AX;  when  rather  I  have  demon- 
strated  already  about  AXM  that  this  would  happen  in 
case  of  the  hypothesis  of  a  common  segment:  Solely 
therefore  I  here  afiirm,  in  one  merely  of  those  new  posi- 
tions,  as  suppose  while  it  fits  XC,  may  be  retained  by  it 
the  same  arbitrary  continuation,  which  it  has  in  the  first 
position,  where  from  the  point  A  it  goes  out  through  X 
toward  the  point  B. 

And  this  indeed  is  demonstrated  thus.  For  first  it  is 
evident  that  the  continuation  AXB  cannot  be  wholly 
similar,  or  equal  to  the  continuation  AXC,  if  each  is 
considered  toward  the  same  part  whether  left  or  right; 
because  otherwise  in  such  position  AXB,  AXC  must 
mutually  coincide ;  which  is  against  the  hypothesis  of  that 
common  segment  AX :  I  say,  must  coincide ;  provided 
that  of  course,  in  relation  to  the  same  fixed  AX,  the  con- 
tinuations  XB,  and  XC  in  the  plane  concerned  extend 
just  similarly  toward  the  same  part  whether  left  or  right. 

Secondly  is  evident  that  nothing  prevents  the  af oresaid 
continuation  AXB,  considered  toward  one  part,  as  suppose, 
toward  the  left,  being  precisely  similar,  or  equal  to  the 
continuation  AXC,  considered  toward  the  opposite  part, 

183 


ram,  adeo  ut  propterea,  sine  ulla  immutatione  in  ipsa 
AXB,  locari  haec  possit  ad  congruendum  in  eodem  plano 
alteri  AXC.  At  manifeste  repugnat,  quod  rursum,  sine 
ulla  immutatione  illius  suae  continuationis,  locari  ea  pos- 
sit  in  eodem  plano  ad  congruendum  alteri  AXM,  quae  ni- 
mirum  dividat  in  X  illum  qualemcunque  angulum  BXC. 
Quod  enim  continuatio  AXB  alia  plane  sit  a  continuatione 
AXM,  si  utraque  consideretur  versus  eandem  seu  lae-[76] 
vam,  seu  dexteram  partem,  ex  eo  manifestum  esse  debet; 
quia  caeterum  (ut  in  simili  observatum  jam  est)  in  ea  tali 
positione  deberent  invicem  congruere  ipsae  AXB,  AXM. 
Sed  neque  sustineri  potest,  quod  continuatio  AXB  versus 
unam  partem,  ut  puta  ad  laevam,  similis  plane  sit,  aut 
aequalis  continuationi  AXM  versus  partem  adversam,  ut 
puta  ad  dexteram;  quia  caeterum  continuatio  AXM  ver- 
sus  dexteram  similis  plane  foret,  aut  aequalis  continua- 
tioni  AXC  versus  eandem  dexteram  partem  propter  sup- 
positam  omnimodam  similitudinem,  aut  aequalitatem  inter 
modo  dictam  continuationem,  et  illam  aliam  AXB  ver- 
sus  laevam.  Tunc  autem  in  ea  tali  positione  (ut  est  prae- 
dictum)  deberent  invicem  congruere  ipsae  AXM,  AXC; 
quod  est  contra  praesentem  hypothesim. 

Ex  quibus  omnibus  infero:  eam  suppositam  rectam 
AXB  (quatenus  quidem  intelhgatur  conservare  suam  il- 
lam  qualemcunque  continuationem  ex  puncto  A  versus 
B)  recipere  non  posseduphcem  aham  in  eodem  plano  posi- 
tionem,  in  quarum  utraque  portio  quidem  AX  in  eodem 
situ  persistat,  portio  vero  altera  XB  in  una  iharum  dua- 
rum  positionum  congruat  (exemph  causa)  ipsi  XC,  et 
in  aha  positione  congruat  ipsi  XM.  Quod  erat  propo- 
situm. 

Dico  tertio:  eandem  suppositam  rectam  AXB  non 


"»4 


as  suppose,  toward  the  right,  so  that  consequently,  with- 
out  any  change  in  AXB,  this  may  be  brought  to  con- 
gruence  with  the  other  AXC  in  the  same  plane. 

But  it  is  manifestly  contradictory,  that  on  the  other 
hand,  without  any  change  of  its  prolongation,  this  can  be 
brought  in  the  same  plane  into  congruence  with  the  other 
AXM,  which  indeed  at  X  divides  that  arbitrary  angle 
BXC. 

For  that  the  prolongation  AXB  is  plainly  other  than 
the  prolongation  AXM,  if  each  is  considered  toward  the 
same  part,  whether  left  [76]  or  right,  must  be  manifest 
from  this;  because  otherwise  (as  already  observed  in 
Hke  case)  in  such  a  situation  AXB,  AXM  must  mutu- 
ally  fit. 

But  neither  can  it  be  maintained,  that  the  prolonga- 
tion  AXB  toward  one  part,  as  suppose  toward  the  left, 
is  wholly  similar,  or  equal  to  the  prolongation  AXM 
toward  the  opposite  part,  as  suppose  toward  the  right; 
because  otherwise  the  prolongation  AXM  toward  the 
right  would  plainly  be  similar,  or  equal  to  the  prolonga- 
tion  AXC  toward  the  same  right  side,  because  of  the 
assumed  complete  similitude,  or  equaHty  between  the  just 
cited  prolongation,  and  that  other  AXB  toward  the  left. 

But  then  in  such  a  situation  (as  previously  remarked) 
AXM,  AXC  should  mutually  fit;  which  is  against  the 
present  hypothesis. 

From  all  which,  I  infer:  the  assumed  straight  AXB 
(in  so  far  as  it  is  understood  to  retain  its  arbitrary  pro- 
longation  from  the  point  A  toward  B)  cannot  have  two 
different  positions  in  the  same  plane,  in  both  of  which 
the  portion  indeed  AX  remains  in  the  same  location,  but 
the  other  portion  XB  in  one  of  those  two  positions  fits 
(for  example)  XC,  and  in  the  other  position  iits  XM. 

Quod  erat  propositum. 

I  say  thirdly:  the  assumed  straight  AXB  can  in  no 

185 


alia  ratione  conservare  posse  suam  illam  qualemcunque 
continuationem,  dum  ejusdem  portio  XB  intelligitur  trans- 
ferri  per  nova,  et  nova  loca  usque  ad  congruendum  in  illo 
quodam  plano  ipsi  XC,  persistente  interim  in  eodem  suo 
loco  portione  AX;  non  posse,  inquam,  conservare  suam 
illam  qualemcunque  continuationem,  nisi  quatenus  portio 
ipsa  XB  intelligatur  ascendere,  aut  descendere  ad  existen- 
dum  cum  illa  immota  AX  in  novis,  et  novis  planis,  usque 
dum  redeat  ad  antiquum  planum,  congruens  ibi  praedictae 
XC.[77] 

Id  enim  censeri  potest  jam  demonstratum ;  quia  scili- 
cet  nulla  alia  in  eodem  illo  plano  reperiri  potest  positio, 
juxta  quam  ipsa  AXB  (persistente  portione  AX  in  suo 
eodem  loco)  conservet  suam  illam  qualemcunque  conti- 
nuationem,  praeterquam  ubi  deveniat  ad  congruendum 
praedictae  AXC. 

Dico  quarto :  designari  posse  in  eo  arcu  BC  tale  punc- 
tum  D,  ad  quod  si  jungatur  XD,  jam  ipsa  AXD  non 
modo  recta  linea  sit,  sed  rursum  ita  se  habeat,  ut  conti- 
nuatio  AXD,  considerata  versus  laevam,  aequalis  plane 
sit,  aut  similis  eidem  continuationi  consideratae  versus 
dexteram. 

Demonstratur.  Et  prior  quidem  pars  (qualecunque 
sit  illud  punctum  D  in  arcu  BC  designatum)  eo  modo 
ostenditur,  quo  supra  usi  sumus  circa  continuatam  AXM. 
Posterior  vero  pars  ita  evincitur.  Nam  hic  supponimus 
duas  rectas  AXB,  AXC,  sub  eodem  communi  segmento 
AX.  Praeterea  supponimus  continuationem  AXB  versus 
laevam  non  esse  omnino  similem,  aut  aequalem  eidemmet 
continuationi  versus  dexteram;  quia  stante  omnimoda 
ejusmodi  similitudine,  aut  aequalitate,  facile  ostenditur 
nulli  alteri  rectae  lineae  commune  esse  posse  illud  segmen- 
tum  AX,  prout  nempe  sic  demonstrabimus  de  illa  conti- 
nuata  AXD.  Tandem  consequenter  supponimus  conti- 
nuatam  illam  AXB  ita  locari  posse  in  eodem  plano,  ut 

i86 


other  way  retain  its  arbitrary  prolongation,  while  its  part 
XB  is  supposed  to  be  transferred  through  new  and  new 
positions  even  to  fitting  XC  in  that  one  plane,  the  portion 
AX  remaining  meanwhile  in  the  same  place;  I  say  it 
cannot  retain  its  chosen  continuation,  except  in  so  far 
as  the  portion  XB  is  understood  to  ascend,  or  to  descend 
to  be  with  the  fixed  AX  in  new,  and  new  planes,  until 
it  returns  to  the  old  plane,  fitting  there  the  aforesaid 

xc.  \.m 

For  this  may  be  adjudged  already  demonstrated ;  be- 
cause  obviously  no  position  in  that  same  plane  can  be  f ound, 
at  which  AXB  (the  portion  AX  remaining  in  its  place) 
retains  its  chosen  prolongation,  except  where  it  comes  to 
congruence  with  the  aforesaid  AXC. 

I  say  fourthly :  in  the  arc  BC  such  a  point  D  can  be 
designated  that,  if  XD  be  joined,  then  this  AXD  not  only 
is  a  straight  line,  but  moreover  it  lies  so,  that  the  pro- 
longation  AXD,  considered  toward  the  left,  is  wholly 
equal,  or  similar  to  the  same  prolongation  considered 
toward  the  right. 

Proof.  The  first  part  (whatever  be  the  point  D 
designated  in  the  arc  BC)  is  shown  by  the  method  used 
above  in  regard  to  the  prolongation  AXM. 

But  the  second  part  is  proved  thus.  We  suppose  here 
two  straights  AXB,  AXC  with  the  same  common  segment 
AX.  Further  we  suppose  the  prolongation  AXB  toward  the 
left  not  to  be  wholly  similar,  or  equal  to  the  same  prolonga- 
tion  toward  the  right ;  because,  such  a  complete  similitude 
or  equality  holding  good,  it  is  easily  shown  that  segment 
AX  can  be  common  to  no  other  straight  line,  just  as  we 
shall  demonstrate  of  the  prolongation  AXD.  Finally  in 
consequence  we  suppose  the  prolongation  AXB  may  so 


It7 


sub  eodem  immoto  segmento  AX  congruat  cuidam  alteri 
AXC,  in  qua  nimirum  continuatio  ipsa  AXC  versus  dex- 
teram  similis  plane  sit,  aut  aequalis  continuationi  AXB 
versus  laevam,  ac  rursum  continuatio  AXC  versus  laevam 
similis  plane  sit,  aut  aequalis  continuationi  AXB  versus 
dexteram. 

His  stantibus :  si  ad  quodvis  punctum  M  sumptum  in 
eo  arcu  BC  jungatur  XM ;  vel  continuatio  AXM  erit  U8] 
sibi  ipsi  plane  uniformis  relate  ad  laevam,  ac  dexteram 
partem  ipsius  AX ;  vel  non.  Si  primum ;  demonstrabo  de 
ista  AXM,  quod  statim  demonstraturus  sum  de  illa  con- 
tinuata  AXD.  Si  secundum,  ergo  praedicta  AXM  ita 
rursum  locari  poterit  in  eodem  plano,  ut  sub  eodem  im- 
moto  segmento  AX  congruat  cuidam  alteri  AXF,  in  qua 
nimirum  continuatio  ipsa  AXF  versus  dexteram  similis 
plane  sit,  aut  aequalis  continuationi  AXM  versus  laevam, 
ac  rursum  continuatio  AXF  versus  laevam  similis  plane 
sit,  aut  aequalis  continuationi  AXM  versus  dexteram. 
Porro,  cum  punctum  M  supponi  possit  vicinius  puncto  B, 
quam  punctum  C,  non  cadet  punctum  F  in  ipsum  punctum 
C ;  quia  sic  continuatio  AXM  versus  laevam  similis  plane 
foret,  aut  aequalis  continuationi  AXF,  sive  AXC  versus 
dexteram,  ac  propterea  similis  plane,  aut  aequalis  conti- 
nuationi  AXB  versus  laevam,  quod  est  absurdum,  cum  il- 
lae  duae  XM,  XB,  non  sibi  invicem  congruant  in  sua 
tali  positione.  Sed  neque  etiam  existet  punctum  F  ultra 
punctum  C  in  eo  arcu  BC  ulterius  producto ;  quia  sic  uni- 
formi  ratiocinio  ostendetur,  contra  hypothesim,  quod 
etiam  punctum  M  deberet  existere  in  eo  arcu  CB  ulterius 
producto,  adeo  ut  nimirum  ipsa  XM  divideret  versus  lae- 
vam  eum  qualemcunque  angukim  AXB,  prout  XF  pone- 
retur  dividere  versus  dexteram  eum  qualemcunque  angu- 
lum  AXC:  Deberet,  inquam,  sic  existere,  ad  eum  utique 


i88 


be  located  in  that  plane,  that  with  its  fixed  segment  AX 
it  fits  a  certain  other,  AXC,  in  so  far  as  truly  the  pro- 
longation  AXC  toward  the  right  is  exactly  similar,  or 
equal  to  the  prolongation  AXB  toward  the  left,  and  more- 
over  the  prolongation  AXC  toward  the  left  is  precisely 
similar,  or  equal  to  the  prolongation  AXB  toward  the  right. 

This  remaining:  if,  assuming  any  point  M  in  the  arc 
BC,  we  join  XM;  either  the  prolongation  AXM  will  be 
[78]  precisely  uniform  in  relation  to  the  left,  and  the 
right  side  of  AX;  or  not.  If  the  first;  I  shall  demon- 
strate  of  AXM,  what  immediately  I  shall  have  demon- 
strated  of  the  prolongation  AXD.  If  the  second,  there- 
fore  the  aforesaid  AXM  can  in  turn  be  so  located  in  the 
same  plane,  that  with  the  same  fixed  segment  AX  it  fits 
a  certain  other  AXF,  in  which  truly  the  prolongation 
AXF  toward  the  right  is  precisely  similar,  or  equal  to 
the  prolongation  AXM  toward  the  left,  and  moreover  the 
continuation  AXF  toward  the  left  is  precisely  similar,  or 
equal  to  the  prolongation  AXM  toward  the  right. 

Furthermore,  since  the  point  M  may  be  supposed 
nearer  to  the  point  B  than  is  the  point  C,  the  point  F  does 
not  fall  upon  the  point  C;  because  thus  the  prolongation 
AXM  toward  the  left  would  be  precisely  similar,  or 
equal  to  the  prolongation  AXF,  or  AXC  toward  the 
right,  and  therefore  precisely  similar,  or  equal  to  the 
prolongation  AXB  toward  the  left,  which  is  absurd,  since 
the  two  XM,  XB  do  not  mutually  fit  each  other  in  such 
position  of  theirs. 

But  neither  also  is  the  point  F  beyond  the  point  C  in 
the  arc  BC  produced  farther  on;  because  thus  by  like 
reasoning  is  shown,  against  the  hypothesis,  that  also  the 
point  M  must  be  in  the  arc  CB  produced  farther  on,  so 
that  XM  would  divide  toward  the  left  the  arbitrary  angle 
AXB,  just  as  XF  would  be  posited  to  divide  toward  the 
right  the  arbitrary  angle  AXC :  I  say  must  so  lie,  to  the 

189 


finem,  ut  ea  AXM  sub  eodem  immoto  segmento  AX  lo- 
cari  rursum  possit  in  eodem  plano  ad  congruendum  illi 
alteri  AXF,  in  qua  nimirum  continuatio  ipsa  AXF  ver- 
sus  dexteram  similis  plane  sit,  aut  aequalis  continuationi 
AXM  versus  laevam,  ac  rursum  continuatio  AXF  versus 
laevam  similis  plane  sit,  aut  aequalis  continuationi  AXM 
versus  dexteram. 

Quoniam  vero  arcus  BC  major  est  ejusdem  portione 
[79]  MF,  designarique  uniformiter  possunt  in  ea  portione 
MF  alia  duo  puncta  cum  minore,  sine  ullo  certo  termino, 
intercapedine ;  alterutrum  sane  in  hac  praedictorum  punc- 
torum  approximatione  contingere  debet.  Unum  est,  si 
tandem  incidatur  in  unum  idemque  intermedium  punctum 
D,  ad  quod  si  jungatur  XD,  talis  habeatur  continuatio 
AXD,  cui  soH  conveniat  (facta  comparatione  inter  lae- 
vam,  ac  dexteram  partem)  esse  sibi  ipsi  omnino  similem, 
aut  aequalem.  Alterum  est,  si  duo  taHa  inveniantur  dis- 
tincta  puncta  M,  et  F,  ad  quae  junctae  XM,  et  XF,  duas 
exhibeant  continuationes,  unam  AXM,  et  alteram  AXF, 
quarum  utraque  sit  sibi  ipsi,  modo  jam  expHcato,  omnino 
simiHs,  aut  aequaHs.  Hoc  autem  secundum  impossibile 
esse  sic  demonstro.  Nam  ex  ipsis  terminis  constare  potest, 
quod  recta  Hnea,  ex  puncto  A  per  X  ulterius  producta, 
unicam  tantum  sortiri  potest  in  eo  taH  plano  positionem, 
dum  sciHcet  quaedam  superaddita  XF  aeque  omnino  se 
habeat  in  laevam,  et  in  dexteram  partem  praesuppositae 
AX,  seu  non  magis  in  laevam,  quam  in  dexteram  ejusdem 
partem  convergat.  Non  ergo  aHa  erit  continuatio  AXM, 
quae  rursum  aeque  omnino  se  habeat  in  laevam,  et  in  dex- 
teram  partem  ejusdem  AX.  SciHcet  constat  subsistere 
simul  non  posse;  et  quod  continuatio  AXF  versus  dex- 
teram  simiHs  plane  sit,  aut  aequaHs  sibi  ipsi  consideratae 
versus  laevam;  et  quod  alia  quaedam  continuatio  AXM 


190 


end,  that  AXM  with  its  fixed  segment  AX  can  again  be 
50  placed  in  that  plane  as  to  fit  the  other,  AXF  in  so  far 
as  truly  the  prolongation  AXF  toward  the  right  is  pre- 
cisely  similar,  or  equal  to  the  prolongation  AXM  toward 
the  left,  and  moreover  the  prolongation  AXF  toward 
the  left  is  precisely  similar,  or  equal  to  the  prolongation 
AXM  toward  the  right. 

But  since  the  arc  BC  is  greater  than  its  part  [79]  MF, 
and  in  this  portion  MF  in  like  way  may  be  designated 
two  other  points  with  an  interval  less,  without  any  certain 
limit;  truly  one  of  two  things  must  happen  in  this  ap- 
proximation  of  the  aforesaid  points. 

One  is,  if  at  length  is  attained  one  and  the  same 
intermediate  point  D,  to  which  if  XD  is  joined,  such  a 
prolongation  AXD  is  obtained,  as  alone  is  such  as  to  be 
wholly  similar,  or  equal  to  itself  (comparison  made  be- 
tween  the  left  and  the  right  side). 

The  other  is,  if  two  such  distinct  points  M,  and  F  are 
found,  to  which  XM,  and  XF  being  joined,  two  prolonga- 
tions  arise,  one  AXM,  and  the  other  AXF,  of  which  each 
is,  in  the  way  just  explained,  wholly  similar,  or  equal. 

But  this  second  I  prove  to  be  impossible  thus.  For 
f  rom  the  very  terms  can  be  established,  that  a  straight  line 
produced  from  the  point  A  on  through  X,  can  take  in 
the  plane  only  a  single  position,  whilst  obviously  the 
superadded  XF  lies  altogether  equally  toward  the  left, 
and  toward  the  right  side  of  the  assumed  AX,  or  deviates 
not  more  toward  the  left,  than  toward  the  right  side  of 
it.  Therefore  there  will  not  be  another  prolongation 
AXM,  which  also  lies  altogether  equally  toward  the  left, 
and  toward  the  right  of  this  AX. 

Obviously  it  holds  that  it  cannot  happen  at  the  same 
time,  both  that  the  prolongation  AXF  toward  the  right 
is  wholly  similar,  or  equal  to  itself  considered  toward 
the  left,  and  that  another  prolongation  AXM  toward  the 

191 


versus  laevam  (quae,  ex  ipsa  positione,  minor  sit  continua- 
tione  AXF  versus  eandem  laevam)  aequalis  iterum  sit 
eidem  continuationi  versus  dexteram,  quae  certe,  ex  ipsa 
rursum  positione,  major  est  praedicta  continuatione  AXF 
versus  eandem  dexteram. 

Non  ergo  in  eo  arcu  BC  duo  talia  inveniri  possunt 
puncta  M,  et  F,  ad  quae  junctae  XM,  et  XF,  duas  exhibe- 
ant  continuationes,  unam  AXM,  et  alteram  AXF,  qua-[80] 
rum  utraque  sit  sibi  ipsi,  modo  jam  explicato,  omnino 
similis,  aut  aequalis.  Unde  tandem  consequitur  incidi  ali- 
quando  debere  in  unum,  idemque  punctum  D,  ad  quod 
juncta  XD  talem  exhibeat  continuationem  AXD,  cui  soli 
conveniat  (facta  comparatione  inter  laevam,  ac  dexteram 
partem)  esse  sibi  ipsi  omnino  similem,  aut  aequalem. 
Quod  erat  hoc  loco  demonstrandum. 

Dico  tandem  quinto:  eam  solam  AXD  fore  Hneam 
rectam,  nimirum  ex  A  per  X  directe  continuatam  in  D. 
Quamvis  enim  ly  ex  aequo,  in  definitione  lineae  rectae, 
appHcari  primitus  debeat  punctis  intermediis  relate  ad 
puncta  ipsius  extrema;  unde  utique  jam  eHcuimus,  duas 
lineas  rectas  non  claudere  spatium ;  intehigi  tamen  etiam 
debet  de  ejusdem  rectae  Hneae  continuatione  in  directum. 
Itaque  ea  sola  XD  (in  eodem  cum  AX  plano  existens) 
dicetur  esse  continuatio  recta,  sive  in  rectum  praedictae 
AX,  quando  ipsa  neque  in  laevam,  neque  in  dexteram 
iHius  partem  convergat,  sed  utrinque  ex  aequo  procedat; 
adeo  ut  nempe  continuatio  iHa  AXD  versus  laevam  simiHs 
plane  sit,  aut  aequaHs  eidem  continuationi  consideratae 
versus  dexteram.  Inde  enim  fiet,  ut  ihi  soH  AXD  con- 
veniat  non  posse  ab  ea  suscipi  in  eo  taH  plano  aHam  posi- 
tionem  sub  iHa  immota  AX;  cum  certe  (ex  jam  demon- 


19« 


left  (which,  from  its  very  position,  is  less  than  the  pro- 
longation  AXF  toward  the  same  left)  again  is  equal  to 
the  same  continuation  toward  the  right,  which  truly, 
again  from  its  very  position,  is  greater  than  the  afore- 
said  prolongation  AXF  toward  the  same  right. 

Therefore  in  the  arc  BC  cannot  be  found  two  such 
points  M,  and  F,  that  the  joins  XM,  and  XF,  present 
two  prolongations,  one  AXM,  and  the  other  AXF,  of 
which  [80]  each  is  to  itself,  in  the  way  just  explained, 
wholly  similar,  or  equal. 

Whence  at  length  follows,  that  somewhere  must  be 
attained  one  and  the  same  point  D,  to  which  the  join  XD 
presents  such  a  prolongation  AXD,  that  to  it  alone  be- 
longs  to  be  wholly  similar,  or  equal  to  itself  (comparison 
made  between  left,  and  right  side). 

Quod  erat  hoc  loco  demonstrandum. 

At  length  I  say  fifthly :  this  AXD  alone  is  a  straight 
line,  namely  from  A  through  X  directly  continued  on 
to  D. 

For  though  the  phrase  ex  aequo,  in  the  definition  of 
the  straight  line,  should  primarily  be  applied  to  points 
intermediate  in  relation  to  its  extreme  points;  whence  in 
particular  we  have  just  deduced,  two  straight  lines  do 
not  inclose  a  space;  nevertheless  it  should  also  be  under- 
stood  of  the  direct  prolongation  of  this  straight  line. 

Therefore  alone  this  AD  (lying  in  the  same  plane 
with  AX)  is  said  to  be  the  sfraight  prolongation  (or 
in  a  straight)  of  the  aforesaid  AX,  when  that  deviates 
neither  toward  the  left,  nor  toward  the  right  side  of  it, 
but  from  each  side  proceeds  ex  aequo;  so  that  the  pro- 
longation  AXD  is  toward  the  left  clearly  similar,  or 
equal  to  the  same  prolongation  considered  toward  the 
right.  For  thence  it  will  follow,  that  alone  to  AXD 
pertains  that  another  position  cannot  be  taken  by  it  \n 
the  plane,  while  AX  is  fixed;  when  truly  (from  what 


stratis)  illae  aliae  AXB,  et  AXM,  citra  omnem  suarum 
talium  continuationum  immutationem,  suscipere  possint 
sub  eadem  immota  AX  alias  in  eodem  plano  positiones, 
quales  sunt  ipsarum  AXC,  et  AXF.  Igitur  illa  sola  AXD, 
cujus  nempe  continuatio  XD  tum  in  eodem  cum  ipsa  AX 
plano  existat,  tum  etiam  aeque  omnino  se  habeat  in  lae- 
vam,  ac  dexteram  partem  praedictae  AX,  est  linea  recta 
juxta  explicatam  definitionem,  seu  continuatio  in  rectum 
ejusdem  praesuppositae  rectae  AX. 

Ex  quibus  omnibus  tandem  constat  evenire  non  posse, 
ut  unum  quodpiam  sit  commune  segmentum  duarum  [81] 
rectarum.     Quod  erat  demonstrandum. 

COROLLARIUM. 

Ex  duobus  praemissis  Lemmatis  tria  opportune  sub- 
notare  licet.  Unum  est:  duas  rectas,  neque  sub  infinite 
parva  inter  ipsas  distantia,  claudere  spatium  posse.  Ratio 
est,  quia  (prout  in  primo  Lemmate)  vel  utraque  illarum 
sub  duobus  illis  communibus  extremis  punctis  immotis 
revolvi  posset  ad  novum  situm  occupandum,  et  sic  (ex 
jam  tradita  lineae  rectae  definitione)  neutra  foret  linea 
recta:  vel  una  tantum  in  suo  eodem  situ  persisteret,  et 
sic  illa  sola  recta  linea  foret.  Quod  autem  nequeat  utra- 
que  in  eodem  ipso  situ  persistere,  dum  aliquod  conclu- 
dant  spatium,  etiamsi  infinite  parvum,  manifestum  fiet 
consideranti  posse  facieni  illius  plani,  in  quo  illae  duae 
consistunt,  converti  de  superna  in  infernam,  manentibus 
caeteroquin  in  suo  eodem  loco  duobus  illis  extremis 
punctis. 

Alterum  est:  neque  item  ullam  lineam  rectam,  in 
quantalibet  ejusdem  productione  in  directum,  diffindi 
posse  in  duas,  quamvis  sub  infinite  parva  intercapedine. 
Ratio  est;  quia  (prout  in  praecedente  Lemmate)  conti- 


194 


has  just  now  been  proved)  those  others,  AXB,  and  AXM, 
without  any  change  of  their  prolongations,  can,  with  the 
same  fixed  AX,  take  other  positions  in  the  same  plane, 
such  as  AXC  and  AXF. 

Therefore  alone  AXD,  whose  prolongation  XD  not 
only  is  in  the  same  plane  with  AX,  but  also  Hes  alto- 
gether  in  Hke  manner  toward  the  left,  and  the  right  side 
of  the  aforesaid  AX,  is  a  straight  Hne  in  accordance  with 
the  discussed  definition,  or  the  prolongation  in  a  straight 
of  the  assumed  straight  AX. 

From  all  which  finany  is  estabHshed  as  impossible, 
that  one  segment  can  be  common  to  two  [81]  straight  Hnes. 

Quod  erat  demonstrandum. 

COROLLARY. 

From  the  two  preceding  lemmata  three  things  may 
opportunely  be  noted. 

One  is :  not  even  with  an  infinitely  small  distance  be- 
tween  them  can  two  straights  inclose  a  space. 

The  reason  is,  because  (just  as  in  Lemma  I)  either 
each  of  them  with  the  two  common  extreme  points  fixed 
can  be  revolved  into  occupying  a  new  position,  and  so 
(from  the  definition  of  the  straight  Hne  already  given) 
neither  will  be  a  straight  Hne :  or  only  one  remains  in  the 
same  place,  and  so  it  alone  is  a  straight  Hne. 

But  that  both  cannot  remain  in  the  same  place,  while 
they  inclose  any  space,  even  if  infinitely  Httle,  wiH  be 
manifest  from  considering  that  a  face  of  the  plane,  in 
which  the  two  are,  can  be  converted  from  upper  to  lower, 
the  two  extreme  points  withal  remaining  in  the  same 
place. 

Another  is :  nor  moreover  can  any  straight  Hne,  in 
any  direct  production  of  it,  spHt  into  two,  although  with 
an  interval  infinitely  smah. 

The   reason   is,   because    (just   as   in   the   preceding 

195 


nuatio  in  directum  praesuppositae  cujusdam  simplicis  rec- 
tae  AX  non  alia  esse  intelligitur  praeter  unam  XD,  quae 
ex  aequo  utrinque  procedat  relate  ad  laevam,  ac  dexteram 
partem  praedictae  AX;  ex  quo  utique  fiat,  ut  sub  ea  ini- 
mota  AX  non  aliam  ipsa  immutata  habere  possit  in  eo 
plano  positionem.  Quod  autem  in  eodem  plano  alia  quae- 
dam  ad  laevam  decerni  possit  XM,  infinite  parum  dissi- 
liens  ab  ipsa  XD,  nihil  suffragatur.  Nam  rursum  alia 
item  ad  dexteram  designari  poterit  XF,  quae  uniformiter 
infinite  parum  dissiHat  ab  eadem  XD.  Quare  (prout  in 
praecitato  Lemma-[82]te)  illa  sola  AXD  erit  linea  recta 
a  nobis  definita. 

Tertium  tandem  est:  in  hoc  ipso  secundo  Lemmate 
censeri  posse  immediate  demonstratam  1 .  undecimi ;  quod 
nempe  ejusdem  rectae  nequeat  pars  una  quidem  in  sub- 
jecto  plano  existere,  et  altera  in  sublimi. 

LEMMA  III. 

Si  duae  rectae  AB,  CXD  sibi  invicem  occurrant  (fig.  39.) 

in  aliquo  ipsarum  intermedio  puncto  X,  non  ibi  se  in- 

vicem  contingent,  sed  una  alteram  ibidem  secabit. 

Demonstratur.  Si  enim  fieri  potest,  tota  CXD  ad 
unam  eandemque  partem  ipsius  AB  consistat.  Jungatur 
AC.  Non  erit  porro  AC  eadem  cum  ipsa  vekiti  con- 
tinuata  AXC;  quia  caeterum  (contra  praecedens  Lemma) 
duarum  rectarum,  unius  AXC,  et  alterius  praesuppositae 
DXC,  unum  idemque  foret  commune  segmentum  XC. 
Itaque  jungatur  BC.  Non  erit  rursum  haec  BC  continua- 
tio  ipsius  BA  usque  in  punctum  C;  ne  duae  rectae,  una 
XAC,  portio  ipsius  BAC,  et  altera  XC  spatium  claudant, 
contra  praemissum  Lemma  primum.    Igitur  ea  BC  vel  se- 


196 


lemma)  the  direct  prolongation  of  any  assumed  simple 
straight  AX  cannot  be  understood  to  be  other  than  the 
one  XD,  which  proceeds  ex  aequo  on  both  sides  in  rela- 
tion  to  the  left,  and  right  side  of  the  aforesaid  AX; 
from  which  assuredly  follows,  that  with  AX  fixed  it  can- 
not,  itself  unchanged,  have  another  position  in  this  plane. 

But  that  in  the  same  plane  a  certain  other  XM  can 
be  designated  to  the  left,  spHtting  infinitely  Httle  from 
XD,  nothing  avails.  For  again  another,  XF,  Hkewise  to 
the  right  could  be  designated,  which  just  so  spHts  in- 
iinitely  Httle  from  the  same  XD.  Wherefore  (as  in  the 
the  lemma  before  cited)  [82]  alone  AXD  will  be  the 
straight  line  defined  by  us. 

The  third  finally  is :  in  this  second  lemma  may  be 
judged  immediately  demonstrated  Eu.  XI.  1  ;  that  of  the 
same  straight  one  part  cannot  be  in  a  lower  plane,  and 
another  in  an  upper. 

LEMMA  III. 

//  two  straights  AB,  CXD  meet  each  other  (fig.  39)  in 

any  intermediate  point  X  of  theirs,  they  do  not  there 

touch  each  other,  hut  one  cuts  the  other  there. 

Proof.  For  if  that  were  possible,  the  whole  CXD 
lies  on  one  and  the  same  side  of  AB.  Join  AC.  Then 
AC  will  not  be  the  same  with  AXC 
as  if  prolonged;  because  otherwise 
(against  the  preceding  lemma)  of 
two  straights,  one  AXC,  and  the 
other  the  assumed  DXC,  there 
would  be  one  and  the  same  com- 
mon  segment  XC. 

And  so  join  BC.  Again  this  BC  will  not  be  a  pro- 
longation  of  BA  to  the  point  C;  lest  two  straights,  one 
XAC,  portion  of  this  BAC,  and  the  other  XC  inclose  a 
space,  against  the  preceding  Lemma  I. 

197 


cabit  in  aliquo  puncto  L  ipsam  XD,  sive  praesuppositam 
rectam  DXC ;  et  tunc  rursum  duae  rectae  lineae,  una  LC 
portio  ipsius  BC,  et  altera  LXC  portio  praedictae  DXC, 
spatium  claudent ;  vel  alterutrum  extremum  punctum  sive 
A  ipsius  BA,  sive  D  ipsius  CXD,  claudetur  intra  spatium 
comprehensum  ipsis  CX,  XB,  et  alterutra  vel  BFC,  vel 
BHC.  At  in  utroque  casu  idem  absurdum  consequitur: 
Sive  enim  BA  protracta  per  A  occurrat  ipsi  BFC  in  ali- 
quo  puncto  F;  sive  CXD  protracta  per  D  occurrat  ipsi 
BHC  in  aliquo  puncto  H :  in  idem  semper  absurdum  inci- 
dimus,  quod  duae  rectae  spatium  claudant;  nimirum  aut 
[83]  recta  BF  portio  ipsius  BFC  una  cum  altera  BAF ;  aut 
recta  HC,  portio  ipsius  BHC,  una  cum  altera  praesup- 
posita  recta  continuata  CXDH. 

Porro  idem,  aut  majus  absurdum  consequitur,  si  illa 
BA  protracta  per  A  occurrat  in  aliquo  puncto  vel  ipsi  CX, 
vel  sibi  ipsi  in  aliquo  puncto  suae  portionis  XB.  Atque  id 
similiter  valet,  si  altera  CXD  protracta  per  D  occurrat  in 
aliquo  puncto  vel  ipsi  XB,  vel  sibi  ipsi  in  aliquo  puncto 
suae  portionis  CX. 

Itaque  constat,  quod  duae  rectae  AB,  CXD  sibi  invi- 
cem  occurrentes  in  aliquo  ipsarum  intermedio  puncto  X, 
non  ibi  se  invicem  contingent,  sed  una  alteram  ibiden: 
secabit.    Quod  erat  etc. 

LEMMA  IV. 

Omnis  diameter  dividit  bifariam  siium  circulum, 
ejusque  circumferen  tiam . 

Demonstratur.  Esto  circulus  (recole  fig.  23.)  MDH- 
NKM,  cujus  centrum  A,  et  diameter  MN.  Intelligatur 
illius  circuli  portio  MNKM  ita  revolvi  circa  immota 
puncta  M,  et  N,  ut  tandem  accommodetur,  seu  coaptetur 
reliquae  portioni  MNHDM.     Constat  primo  totam  dia- 


198 


Therefore  this  BC  either  will  cut  XD  (or  the  as- 
sumed  straight  DXC)  in  some  point  L;  and  then  again 
two  straight  lines,  one  LC,  portion  of  this  BC,  and  the 
other  LXC,  portion  of  the  aforesaid  DXC,  inclose  a 
space;  or  one  of  the  extreme  points  whether  A  of  BA, 
or  D  of  CXD,  is  inclosed  within  the  space  bounded  by 
CX,  XB,  and  either  BFC,  or  BHC. 

But  in  either  case  the  same  absurdity  follows:  For 
whether  BA  produced  through  A  strikes  BFC  in  a  point 
F;  or  CXD  produced  through  D  strikes  BHC  in  a  point 
H;  always  we  come  upon  the  same  absurdity,  that  two 
straights  inclose  a  space;  forsooth  either  the  straight[83] 
BF  portion  of  BFC  together  with  the  other  BAF ;  or  the 
straight  HC,  portion  of  BHC,  together  with  the  other 
assumed  straight  prolonged  CXDH. 

Furthermore  the  same,  or  a  greater  absurdity  fol- 
lows,  if  BA  produced  through  A  meets  in  any  point 
either  CX,  or  its  own  self  in  any  point  of  its  portion  XB. 

And  this  Hkewise  holds,  if  the  other  CXD  produced 
through  D  meets  in  any  point  either  XB,  or  its  own  self 
in  any  point  of  its  portion  CX. 

Therefore  is  estabhshed,  that  two  straights  AB,  CXD 
meeting  each  other  in  any  intermediate  point  X  of  theirs, 
do  not  there  touch  each  other,  but  one  will  cut  the  other 
there. 

Quod  erat  etc. 

LEMMA  IV. 

Every  diameter  bisecfs  its  circle,  and  the  circumference 

of  it. 

Proof.     Let  there  be  a  circle  (recall  fig.  23)  MDH- 

NKM,  A  its  center,  and  MN  a  diameter.     Of  this  circle 

the  portion  MNKM  is  thought  so  to  revolve  about  the 

fixed  points  M,  and  N,  that  at  length  it  is  superimposed 

upon,  or  appHed  to  the  remaining  portion  MNHDM. 

109 


metrum  MAN  quoad  omnia  ipsius  puncta  in  eodem  situ 
esse  mansuram:  ne  duae  rectae  lineae  (contra  praecedens 
Lemma  primum)  spatium  claudant.  Constat  secundo 
nullum  punctum  K  circumferentiae  NKM  casurum  vel 
intra,  vel  extra  superficiem  clausam  diametro  MAN,  et 
altera  circumferentia  NHDM ;  ne  scilicet  contra  naturam 
circuli,  unus  radius  v.  g.  AK  minor  sit,  aut  major  altero 
ejusdem  circuli  radio  v.  g.  AH.  Constat  tertio  quem- 
libet  radium  MA  continuari  unice  posse  in  rectum  per[84] 
alterum  quendam  radium  AN,  ne  (contra  praecedens 
Lemma  secundum)  duae  suppositae  rectae  lineae,  ut  puta 
MAN,  MAH,  unum  idemque  commune  habeant  segmen- 
tum  MA.  Constat  quarto  (ex  proxime  antecedente  Lem- 
mate)  omnes  cujusvis  circuli  diametros  se  invicem  in  cen- 
tro  secare,  et  ex  nota  natura  circuli  bifariam. 

Ex  quibus  omnibus  constare  potest,  quod  diameter 
MAN  tum  dividit  exactissime  suum  circulum,  ejusque 
circumferentiam  in  duas  aequales  partes,  tum  etiam  as- 
sumi  universim  potest  pro  qualibet  ejusdem  circuli  dia- 
metro.    Quod  erat  etc. 

SCHOLION. 

Hanc  eandem  veritatem  demonstratam  leges  apud 
Clavium  a  Thalete  Milesio,  sed  fortasse  non  exhausta 
omni  qualibet  objectione. 

LEMMA  V. 

Inter  angulos  rectilineos  omnes  anguli  recti  sunt  invicem 

exactissime  aequales,  sine  ullo  defectu  etiam 

infinite  parvo. 

Demonstratur.  Angulum  inter  rectilineos  rectum  de- 
finit  Euclides :    qui  est  aequalis  suo  deinceps.    Non  hunc 


It  is  certain  first  that  as  to  all  its  points  the  whole 
diameter  MAN  will  remain  in  the  same  place;  lest  two 
straight  lines  (against  the  preceding  Lemma  I)  inclose  a 
space. 

It  is  certain  secondly  that  no  point  K  of  the  circum- 
ference  NKM  will  fall  either  within,  or  without  the  sur- 
face  inclosed  by  the  diameter  MAN,  and  the  other  cir- 
cumf erence  NHDM ;  lest  obviously  against  the  nature  of 
the  circle,  one  radius,  f  or  example  AK,  be  less,  or  greater 
than  another  radius  of  the  same  circle,  for  example  AH. 

It  is  certain  thirdly  that  any  radius  MA  can  alone  be 
prolonged  in  a  straight  Hne  by  [84]  a  certain  other  radius 
AN,  lest  (against  the  preceding  Lemma  II)  two  lines 
assumed  straight,  as  suppose  MAN,  MAH,  should  have 
one  and  the  same  common  segment  MA. 

It  is  certain  f  ourthly  (  f  rom  the  immediately  preceding 
lemma)  that  all  the  diameters  of  the  circle  cut  one  an- 
other  in  the  center,  and  from  the  known  nature  of  the 
circle  bisect. 

From  all  which  can  be  estabHshed,  that  not  only  the 
diameter  MAN  most  exactly  divides  its  circle,  and  the 
circumference  of  it  into  two  equal  parts,  but  also  that 
this  may  be  assumed  universally  for  any  diameter  of  this 
circle.  Quod  erat  etc. 

SCHOLION. 
We  read  in  Clavius  that  this  truth  was  demonstrated 
by  Thales  of  Miletus,  but  perhaps  not  to  the  exhausting 
of  every  objection. 

LEMMA  V. 

Among  rectilinear  angles,  all  right  angles  are  exactly 

equal  to  one  another,  without  any  devia- 

tion  even  infinitely  small, 

Proof.     EucHd  defines  a  rectiHnear  angle  as  right: 

which  is  equal  to  its  adjacent.    This  he  does  not  postu- 


postulat  ipse  sibi  concedi,  sed  problematice  demonstrat  in 
sua  Prop.  XI.  Libri  primi.  Ibi  enim  ex  dato  in  recta  BC 
quolibet  puncto  A  (fig.  40.)  docet  excitare  perpendicu- 
larem  AD,  ad  quam  anguli  DAB,  DAC  sint  invicem 
aequales.  Porro  illos  duos  angulos  esse  invicem  exac- 
tissime  aequales,  sine  ullo  defectu  etiam  infinite  parvo, 
constare  potest  ex  Corollario  post  duo  priora  praemissa 
Lemmata  [85]  si  nempe  ipsae  AB,  AC  designatae  sint 
exactissime  aequales. 

Sed  aliqua  oriri  potest  dubitatio,  si  duo  alii  ad  quan- 
dam  alteram  FM  recti  anguli  LHF,  LHM  (fig.  4L)  con- 
ferantur  cum  praedictis  rectis  angulis  DAB,  DAC.  Ita- 
que  HL  aequalis  sit  ipsi  AD,  ac  rursum  posterior  integra 
Figura  ita  intelligatur  superponi  priori,  ut  punctum  H 
cadat  super  punctum  A,  et  punctum  L  super  punctum  D. 
Jam  sic  progredior.  Et  prima  quidem  (ex  praecedente 
Lemmate)  ipsa  FHM  non  praecise  continget  alteram  BC 
in  eo  puncto  A.  Ergo  vel  adamussim  procurret  super  illa 
BC,  vel  eandem  ita  secabit,  ut  unum  ejus  punctum  ex- 
tremum  v.  g.  F  cadat  supra,  et  alterum  M  deorsum.  Si 
primum:  jam  clare  habemus  exactissimam  inter  omnes 
rectilineos  angulos  rectos  aequalitatem  intentam.  At  non 
secundum;  quia  sic  angulus  LHF,  hoc  est  DAF,  minor 
foret  angulo  DAB,  ejusque  supposito  exactissime  aequali 


late  as  conceded  to  him,  but  demonstrates  through  a 
problem  in  his  Bk.  I.  P.  11,  For  there  he  teaches  from  any 
given  point  A  (fig.  40)  in  the  straight  BC  to  erect  a 
perpendicular  AD  at  which  the  angles  DAB,  DAC  are 
equal  to  each  other. 

Moreover  that  those  two  angles  are  precisely  equal 
to  each  other,  without  any  difference  even  infinitely  small, 
follows  from  the  corollary  after  the  first  two  premised 
lemmata,  [85]  if  AB,  AC  are  taken  exactly  equal. 


\^ 


M 


Fig.  40. 


rt 

Fig.  41. 


M 


But  some  doubt  may  arise,  if  two  other  right  angles 
LHF,  LHM  (fig.  41)  at  any  other  straight  FM  are 
compared  with  the  aforesaid  right  angles  DAB,  DAC. 

Therefore  let  HL  be  equal  to  AD,  and  then  the  whole 
latter  figure  is  thought  to  be  superposed  upon  the  former 
so,  that  point  H  falls  upon  point  A,  and  point  L  upon 
point  D. 

Now  I  proceed  thus. 

And  first  indeed  (from  a  preceding  lemma  [HI] )  this 
FHM  does  not  exactly  touch  the  other  BC  in  the  point 
A.  Therefore  either  it  runs  forward  precisely  upon  BC, 
or  will  cut  it  so  that  one  of  its  end  points  for  example  F 
falls  above,  and  the  other  M  below. 

In  the  first  case :  now  clearly  we  have  the  exact  equal- 
ity  asserted  between  all  rectiHnear  right  angles. 

But  not  in  the  second;  because  thus  the  angle  LHF, 
here  it  is  DAF,  will  be  less  than  the  angle  DAB,  and  its 


203 


DAC,  et  sic  multo  minor  angulo  DAM,  sive  LHM ;  con- 
tra  hypothesin.  Deinde  vero  nihil  suffragatur,  quod  an- 
gulus  DAF  infinite  parum  deficiat  ab  angulo  DAB,  sive 
ejus  exactissime  aequaH  DAC,  qui  rursum  solum  infinite 
parum  superetur  ab  angulo  DAM.  Nam  semper  angulus 
DAF,  sive  LHF,  non  erit  exactissime  aequalis  angulo 
DAM,  sive  LHM,  contra  hypothesin. 

Itaque  constat  omnes  rectiHneos  angulos  rectos  esse 
invicem  exactissime  aequales,  sine  uHo  defectu  etiam  in- 
finite  parvo.    Quod  etc.     , 

COROLLARIUM. 

Inde  autem  fit,  ut  quae  ex  uno  dato  cujusvis  rectae 
Hneae  puncto  perpendiculariter  in  aHquo  plano  ad  ean- 
dem  educitur,  ipsa  sit  in  eo  taH  plano  unica  exactissime 
Hnea  recta,  nec  potens  diffindi  in  duas.  [86] 

Post  quinque  praemissa  Lemmata,  eorumque  Corollaria, 

progredi  jam  deheo  ad  demonstrandum  principale 

assumptum  contra  hypothesin  anguli  acuti. 

Ubi  statuere  possum,  tanquam  per  se  notum,  non  mi- 
rius  repugnare,  quod  duae  rectae  Hneae  (sive  ad  finitam, 
sive  ad  infinitam  earundem  productionem)  in  unam  tan- 
dem,  eandemque  rectam  Hneam  coeant;  quam  quod  una 
eademque  Hnea  recta  (sive  ad  finitam,  sive  ad  infinitam 
ejusdem  continuationem)  in  duas  rectas  Hneas  diffindatur, 
contra  praecedens  Lemma  secundum,  ejusque  CoroHa- 
rium.  Quoniam  ergo  naturae  Hneae  rectae  (ex  praece- 
dente  Corol.  proximi  Lemmatis)  oppositum  itidem  est, 
quod  duae  rectae  lineae  ad  unum,  idemque  punctum  cujus- 
dam  tertiae  rectae,  perpendiculares  ipsi  sint  in  eodem  com- 
muni  plano;  agnoscere  oportet  tanquam  absolute  falsam, 
quia  repugnantem  naturae  praedictae,  hypothesin  anguli 
acuti,  juxta  quam  duae  illae  AX,  BX  (fig.  33.)  in  uno 


supposed  exact  equal  DAC,  and  thus  much  less  than  the 
angle  DAM,  or  LHM ;  contrary  to  the  hypothesis. 

Then  it  helps  nothing  that  angle  DAF  differ  infinitely 
little  from  angle  DAB,  or  its  exact  equal  DAC,  which 
again  would  exceed  only  infinitely  Httle  the  angle  DAM. 
For  always  angle  DAF,  or  LHF,  will  not  be  exactly  equal 
to  angle  DAM,  or  LHM,  against  the  hypothesis. 

Therefore  is  established  that  all  rectiHnear  right  an- 
gles  are  exactly  equal  to  one  another,  without  any  differ- 
ence  even  infinitely  small. 

Quod  erat  etc. 

COROLLARY. 

Thence  follows,  that  the  straight  Hne  erected  from  a 
given  point  of  any  straight  perpendicularly  to  it  in  a 
plane,  is,  in  such  plane,  wholly  unique,  nor  can  it  spHt 
in  two.  [86] 

After  the  five  premised  lemmata,  and  their  corollaries,  I 

must  nozv  go  on  to  proof  of  the  principal  objec- 

tion  against  the  hypothesis  of  acute  angle. 

Here  I  may  set  up,  as  known  per  se,  it  is  not  less 
contradictory,  that  two  straight  Hnes  (whether  at  a  finite, 
or  at  an  infinite  prolongation  of  them)  at  length  run 
together  into  one  and  the  same  straight  Hne,  than  that 
one  and  the  same  straight  Hne  (whether  at  a  finite,  or  at 
an  infinite  prolongation  of  it)  splits  into  two  straight 
lines,  against  the  preceding  Lemma  H,  and  its  corollary. 

Since  therefore  it  is  in  like  manner  opposed  to  the 
nature  of  the  straight  line  (from  the  preceding  corollary 
to  the  last  lemma),  that  two  straight  lines  at  one  and  the 
same  point  of  a  third  straight,  be  perpendicular  to  this 
in  the  same  common  plane;  it  is  proper  to  recognize  as 
absolutely  false,  because  repugnant  to  the  aforesaid  na- 
ture,  the  hypothesis  of  acute  angle,  according  to  which 
those  two  AX,  BX  (fig.  33)  in  one  and  the  same  com- 

?05 


eodemque  communi  puncto  X  perpendiculares  esse  debe- 
rent  cuidam  tertiae  rectae,  quae  in  eodem  cum  ipsis  plano 
existeret.    Hoc  autem  erat  principale  demonstrandum. 

SCHOLION. 

Atque  his  subsistere  tutus  possem.  Sed  nullum  non 
movere  lapidem  volo,  ut  inimicam  anguli  acuti  hypothe- 
sim,  a  primis  usque  radicibus  revulsam,  sibi  ipsi  repug- 
nantem  ostendam.  Iste  autem  erit  consequentium  hujus 
Libri  Theorematum  unicus  scopus.  [87] 


io6 


mon  point  X  must  be  perpendicular  to  a  third  straight, 
which  is  in  the  same  plane  with  them. 

Hoc  autem  erat  principale  demonstrandum. 

SCHOLION. 

And  here  I  might  safely  stop.  But  I  do  not  wish  to 
leave  any  stone  unturned,  that  I  may  show  the  hostile 
hypothesis  of  acute  angle,  torn  out  by  the  very  roots, 
contradictory  to  itself. 

However  this  will  be  the  single  aim  of  the  subsequent 
theorems  of  this  Book.  [87] 


«07 


LIBRI  PRIMI  PARS  ALTERA. 

In   qua   idem    Pronunciatum    Euclidaeum    contra 
hypothesin  anguli  acuti  redargutive  demon- 

STRATUR. 

PROPOSITIO   XXXIV. 
In  qua  expenditur  curva  quaedam  enascens  ex  hypothesi 
anguli  acuti. 

Recta  CD  jungat  aequalia  perpendicula  AC,  BD  cui- 
dam  rectae  AB  insistentia.  Tum  divisis  bifariam  in  punc- 
tis  M,  et  H  (fig.  42.)  ipsis  AB,  CD,  jungatur  MH  (ex 
2.  hujus)  utrique  perpendicularis.  Rursum  in  hac  hypo- 
thesi  supponuntur  acuti  anguH  ad  junctam  CD.  Quare 
in  quadrilatero  AMHC  erit  MH  (ex  Cor.  I.  post  3.  hujus) 
minor  ipsa  AC.  Hinc  autem;  si  in  MH  protracta  sumas 
MK  aequalem  ipsi  AC;  puncta  C,  K,  D,  spectabunt  ad 
curvam  hic  expensam.  Deinde  anguH  ad  junctam  CK 
erunt  et  ipsi  (ex  7.  hujus)  acuti.  Igitur  juncta  LX,  quae 
bifariam,  atque  ideo  (ex  2.  hujus)  ad  angulos  rectos  divi- 
dat  ipsas  AM,  CK,  erit  simiHter  (ex  Cor.  I.  post  3  hujus) 
minor  eadem  AC.  Quapropter ;  si  in  LX  protracta  sumas 
LF  aequalem  ipsi  AC,  aut  MK;  etiam  punctum  F  spec- 
tabit  ad  eam  curvam.  Praeterea  jungens  CF,  et  FK  in- 
venies  simiHter  duo  aHa  puncta  ad  eandem  curvam  spec- 
tantia.     Atque  ita  semper.     Quod  autem  dico  pro  inve- 


20S 


PART  11. 


In  which  the  same  Euclidean  postulate  is  demon- 

STRATED  AGAINST  THE  HYPOTHESIS  OF  ACUTE  ANGLE 
BY  REFUTING  THIS. 

PROPOSITION  XXXIV. 

In  which  is  investigated  a  certain  curve  arising  from  the 
hypothesis  of  acute  angle} 

Let  the  straight  CD  join  equal  perpendiculars  AC,  BD 
standing  upon  a  certain  straight  AB.  Then  AB,  CD  being 
bisected  in  the  points  M  and  H  (fig.  42),  MH  is  joined 
perpendicular  (by  P.  H.)  to  each. 
Again  in  this  hypothesis  the  an- 
gles  at  the  join  CD  are  supposed 
acute.  Therefore  in  the  quadri- 
lateral  AMHC  (by  Cor.  I.  to  P 
ni.)  MH  will  be  less  than  AC. 
Hence  now,  if  in  MH  produced 
MK  be  taken  equal  to  AC,  the 
points  C,  K,  D  pertain  to  the 
curve  here  investigated.  Then  the  angles  at  the  join 
CK  will  be  themselves  acute  (by  P.  VII.). 

Therefore  the  join  LX,  which  bisects,  and  therefore 

(by  P.  II.),  is  at  right  angles  to  AM,  CK,  will  be  like- 

wise  (by  Cor.  I.  to  P.  III.)  less  thanAC.  Wherefore,  if 

in  LX  produced  we  assume  LF  equal  to  AC  or  MK,  the 

point  F  also  will  pertain  to  this  curve.    Further,  joining 

CF,  and  FK  we  find  likewise  two  other  points  pertaining 

to  the  same  curve.    And  so  on  forever. 

1  In  the  hypothesis  of  acute  angle  an  equidistant  of  a  straight  has 
its  chords  between  it  and  the  straight. 

209 


I 

r 

hs 

H 

w 


Fig.  42. 


niendis  punctis  inter  puncta  C,  et  K,  idem  etiam  unifor- 
miter  valet  pro  inveniendis  punctis  inter  puncta  K,  et  D, 
scilicet  cur-[88]va  CKD,  enascens  ex  hypothesi  anguli 
acuti,  est  Hnea  jungens  extremitates  omniuni  aequahum 
perpendiculorum  super  eadem  basi  versus  eandeni  partem 
erectorum,  quae  utique  venire  possunt  sub  nomine  recta- 
rum  ordinatim  apphcatarum ;  est,  inquam,  hnea  ejusmodi, 
quae  propter  ipsam,  ex  qua  nascitur,  hypothesim  anguH 
acuti,  semper  est  cava  versus  partes  contrapositae  basis 
AB.  Quod  quidem  hoc  loco  declarandum,  ac  demon- 
strandum  a  nobis  erat. 


PROPOSITIO  XXXV. 

Si  ex  qiwlihet  puncto  L  hasis  AB  ordinatim  applicetur  ad 
eam  curvam  CKD  recta  LF :  Dico  rectam  NFX  per- 
pendicidarem  ipsi  LF  cadere  totam  ex  iitraque  parte 
dehere  versus  partes  convexas  ejusdem  curvae,  atque 
ideo  eam  fore  ejusdem  curvae  tangentem. 

Demonstratur.  Si  enim  fieri  potest,  cadat  quoddam 
punctum  X  (fig.  43.)  ipsius  NFX  intra  cavitatem  ejus- 
dem  curvae.  Demittatur  ex  puncto  X  ad  basim  AB  per- 
pendicularis  XP,  quae  protracta  per  X  occurrat  curvae  in 
quodam  puncto  R.  Jam  sic.  In  quadrilatero  LFXP  non 
erit  anguhis  in  puncto  X  aut  rectus,  aut  obtusus :  Caete- 
rum  (ex  5.  et  6.  hujus)  destrueretur  praesens  hypothesis 
anguh  acuti.  Ergo  praedictus  anguhis  erit  acutus.  Quare 
erit  PX  (ex  Cor.  I.  post  3.  hujus)  et  sic  muko  magis  PR 
major  ipsa  LF.  Hoc  autem  absurdum  est  (ex  praece- 
dente)  contra  naturam  istius  curvae.    Itaque  iha  NF  pro- 


But  what  I  say  for  finding  points  between  the  points 
C  and  K,  the  same  also  holds  good  uniformly  for  finding 
points  between  the  points  K  and  D;  of  course  the  curve 
[88]  CKD,  arising  from  the  hypothesis  of  acute  angle, 
is  the  line  joining  the  extremities  of  all  equal  perpendicu- 
lars  erected  upon  the  same  base  toward  the  same  part, 
which  assuredly  can  come  under  the  name  ordinates ;  it  is, 
I  may  say,  a  line  of  such  sort,  that  on  account  of  the 
hypothesis  of  acute  angle,  from  which  it  arises,  it  always 
is  concave  toward  the  parts  of  the  opposite  base  AB. 

Quod  quidem  hoc  loco  declarandum,  ac  demonstran- 
dum  a  nobis  erat. 


PROPOSITION  XXXV. 
//  from  any  point  L  of  the  hase  AB  the  ordinate  LF  is 
drawn  to  this  curve  CKD :  /  say  the  straight  NFX 
perpendicular  to  LF  must  on  both  sides  fall  wholly 
toward  the  convex  parts  of  this  curve,  and  therefore 
it  will  be  tangent  to  this  curve. 

Proof.  For  if  possible,  let  a  certain  point  X  (fig.  43) 
of  NFX  fall  within  the  cavity 
of  this  curve.  Let  fall  from 
the  point  X  to  the  base  AB  the 
perpendicular  XP,  which  pro- 
longed  through  X  meets  the 
curve  in  a  certain  point  R.  Now 
thus.  In  the  quadrilateral  LFXP 
the  angle  at  the  point  X  will  be 
neither  right  nor  obtuse:  else 
(P.  V.  and  P.  VL)  would  be  destroyed  the  present  hy- 
pothesis  of  acute  angle. 

Therefore  the  aforesaid  angle  will  be  acute.  Where- 
fore  (from  Cor.  I.  to  P.  IIL)  PX  and  so  much  more 
PR  will  be  greater  than  LF.  But  this  is  absurd  (from 
the  preceding)  against  the  nature  of  this  curve. 


n 
Fig.  43. 


tracta  cadere  tota  debet  versus  partes  convexas,  atque  ideo 
ipsa  erit  ejusdem  curvae  tangens.  Quod  erat  demon- 
strandum.  [89] 

PROPOSITIO  XXXVI. 

Si  recta  quaepimn  XF  (fig.  44.)  acutum  angulum  efficiat 
cum  quavis  ordinata  LF,  non  cadet  punctum  X  extra 
cavitatem  curvae,  nisi  prius  ipsa  XF  in  aliquo  puncto 
O  curvam  secuerit. 

Demonstratur.  Constat  sumi  posse  in  ipsa  XF  punc- 
tum  quoddam  X  adeo  vicinum  ipsi  puncto  F,  ut  juncta 
LX  prius  curvam  secet  in  aliquo  puncto  S :  caeterum  ipsa 
XF  vel  non  cadet  tota  extra  cavitatem  curvae,  et  sic  habe- 
mus  intentum;  vel  adeo  non  efficiet  cum  FL  angulum 
acutum,  ut  magis  censenda  jam  sit  in  unicam  rectam  cum 
altera  LF  coire.  Itaque  ex  puncto  S  demittatur  ad  basim 
AB  perpendicularis  SP.  Erit  haec  (ex  34.  hujus)  aequa- 
lis  ipsi  LF.  Est  autem  SP  (ex  19.  prinii)  minor  ipsa 
LS.  Ergo  etiam  LF  minor  est  eadem  LS,  ac  propterea 
multo  minor  ipsa  LX.  Hinc  in  triangulo  LXF  acutus 
erit  angulus  in  puncto  X,  quia  minor  (ex  18.  primi)  an- 
gulo  LFX  supposito  acuto.  Jam  demittatur  ad  FX  per- 
pendicularis  LT.  Cadet  haec  (propter  17.  primi)  ad 
partes  utriusque  anguH  acuti.  Quare  punctum  T  jacebit 
inter  puncta  X,  et  F.  Deinde  ex  puncto  T  demittatur  ad 
basim  AB  perpendicularis  TQ.  Erit  LF  (propter  angulum 


So  NF  produced  must  fall  wholly  toward  the  convex 
parts,  and  so  it  will  be  tangent  to  this  curve. 
Quod  erat  demonstrandum.  [891 

PROPOSITION  XXXVI. 

//  any  straight  XF  (fig.  44)  makes  an  acute  angle  with 
any  ordinate  LF,  the  point  X  does  not  fall  without 
the  cavity  of  the  curve,  unless  previously  XF  has 
cut  the  curve  in  some  point  O. 

pROOF.  It  is  sure  that  some  point  X  may  be  assumed 
in  XF  so  near  to  the  point  F,  that  the  join  LX  previously 
cuts  the  curve  in  some  point  S: 
otherwise  XF  either  does  not  f  all 
wholly  without  the  cavity  of  the 
curve,  and  so  we  have  our  asser- 
tion;  or  so  far  is  it  from  mak- 
ing  with  FL  an  acute  angle,  that 
now  rather  it  must  be  supposed 
to  combine  with  LF  in  one 
straight. 

Accordingly  from  the  point  S  let  fall  to  the  base  AB 
the  perpendicular  SP.  This  will  be  (from  P.  XXXIV.) 
equal  to  LF. 

But  SP  is  (from  Eu.  I.  19)  less  than  LS.  Therefore 
also  LF  is  less  than  LS,  and  consequently  much  less  than 
LX.  Hence  in  triangle  LXF  the  angle  at  point  X  will 
be  acute,  because  less  (from  Eu.  I.  18)  than  the  angle 
LFX  supposed  acute. 

Now  let  f all  to  FX  the  perpendicular  LT.  This  will  f all 
(because  of  Eu.  I.  17)  toward  the  parts  of  each  acute 
angle.  Wherefore  point  T  will  he  between  points  X, 
and  F. 

Then  from  the  point  T  let  fall  to  the  base  AB  the 
perpendicular  TQ.     LF  will  be   (because  of  the  right 

213 


rectum  in  T)  major  ipsa  LT,  et  haec  (propter  angulum 
rectum  in  Q)  major  altera  QT.  Igitur  LF  multo  major 
erit  ipsa  QT.  Hinc  autem;  si  in  QT  protracta  sumatur 
QK  aequalis  ipsi  LF;  punctum  K  (ex  34.  hujus)  ad  prae- 
sentem  curvam  spectabit,  cadetque  idcirco  punctum  T  in- 
tra  cavitatem  ejusdem  curvae.  Non  ergo  recta  FT,  quae 
secat  duas  rectas  QK,  et  LT  in  T,  promoveri  potest  ad 
secandam  protractam  LS  in  puncto  X,  constituto  extra 
cavitatem  praesentis  curvae,  nisi  prius  ipsa  protracta  FT 
secet  in  aliquo  puncto  O  portionem  ejusdem  curvae  inter 
puncta  S,  et  K  [90]  constitutam.  Hoc  autem  erat  demon- 
strandum. 

COROLLARIUM. 

Atque  hinc  manifeste  liquet,  inter  tangentem  hujus 
curvae,  et  ipsam  curvam  locari  non  posse  quandam  rec- 
tam,  quae  tota  ad  hanc,  vel  illam  tangentis  partem  extra 
curvae  cavitatem  cadat;  quandoquidem  recta  sic  locata 
efficere  debet  (ex  praecedente)  angulum  acutum  cum  de- 
missa  ex  puncto  contactus  ad  contrapositam  basim  per- 
pendiculari. 

PROPOSITIO  XXXVII. 

Curva  CKD,  ex  hypothesi  anguli  acuti  enascens,  aequalis 
esse  deheret  contrapositae  basi  AB. 

Ante  demonstrationem  praemitto  sequens  axioma. 

Si  duae  h*neae  bifariam  dividantur,  tum  earum  medie- 
tates,  ac  rursum  quadrantes  bifariam,  atque  ita  in  infini- 
tum  uniformiter  procedatur;  certo  argumento  erit,  duas 
istas  Hneas  esse  inter  se  aequales,  quoties  in  ista  uniformi 
in  infinitum  divisione  comperiatur,  seu  demonstretur,  de- 


;H4 


angle  at  T)  greater  than  LT,  and  this  (because  of  the 
right  angle  at  Q)  will  be  greater  than  QT.  Therefore 
LF  will  be  much  greater  than  QT.  But  hence;  if  in  QT 
produced  QK  is  taken  equal  to  LF;  the  point  K  (from 
P.  XXXIV. )  will  pertain  to  the  present  curve,  and  there- 
fore  point  T  falls  within  the  cavity  of  this  curve. 

Therefore  the  straight  FT,  which  cuts  the  two 
straights  QK,  and  LT  in  T,  cannot  be  extended  to  cut 
LS  prolonged  in  the  point  X,  situated  without  the  cavity 
of  the  present  curve,  unless  previously  the  prolonged  FT 
cuts  in  some  point  O  the  portion  of  this  curve  situated 
between  the  points  S,  and  K.  [90] 

Hoc  autem  erat  demonstrandum. 

COROLLARY. 

And  hence  manifestly  flows,  that  between  the  tangent 
of  this  curve,  and  the  curve  itself  cannot  be  placed  any 
straight,  which,  on  one  or  the  other  side  of  the  tangent 
wholly  falls  without  the  cavity  of  the  curve;  since  a 
straight  so  located  must  (from  the  preceding)  make  an 
acute  angle  with  the  perpendicular  let  fall  f rom  the  point 
of  contact  to  the  opposite  base. 

PROPOSITION  XXXVII. 

The  curve  CKD,  arising  from  the  hypothesis  of  acute 
angle,  must  he  equal  to  the  opposite  base  AB. 

Before  the  demonstration  I  premise  the  following 
axiom. 

If  two  lines  be  bisected,  then  their  halves,  and  again 
their  quarters  bisected,  and  so  the  process  be  continued 
uniformly  in  infinitum ;  it  will  be  safe  to  argue,  those  two 
lines  are  equal  to  each  other,  as  often  as  is  ascertained, 
or  demonstrated  in  that  uniform  division  in  infinitum, 
that  at  length  must  be  attained  two  of  their  mutually 

2T5 


veniri  tandem  debere  ad  duas  illarum  sibi  invicem  respon- 
dentes  partes,  quas  constet  esse  inter  se  aequales. 

Jam  demonstratur  propositum.  Intelligantur  erecta 
ex  basi  AB  ad  eam  curvam  CKD  (fig.  45.)  quotvis  per- 
pendicula  NF,  LF,  PF,  MK,  TF,  VF,  IF;  sintque  aequa- 
les  in  ipsa  basi  AB  portiones  AN,  NL,  LP,  PM,  MT, 
TV,  VI,  IB. 

Constat  primo  angulum  ipsius  AC  cum  ea  curva 
aequalem  fore  singulis  hinc  inde  ad  puncta  F,  sive  ad 
punctum  K,  aut  punctum  D,  praedictarum  perpendicula- 
rium  angulis  cum  eadem  curva.  Si  enim  mistum  quadri- 
late-[91]rum  ANFC  superponi  intelligatur  misto  quadri- 
latero  NLFF,  constituta  basi  AN  super  aequali  basi  NL, 
cadet  AC  super  NF,  et  NF  super  LF,  propter  aequales 
angulos  rectos  ad  puncta  A,  N,  L.  Deinde  propter  aequa- 
litatem  rectarum  (ex  34.  hujus)  AC,  NF,  LF,  cadet 
punctum  C  super  punctum  F  ipsius  NF,  et  hoc  super  alte- 
rum  punctum  F  ipsius  LF.  Praeterea  curva  CF  congruet 
adamussim  ipsi  curvae  FF:  si  enim  una  illarum,  ut  CF 
introrsum,  aut  extrorsum  cadat;  sumpto  quoHbet  puncto 
Q  inter  puncta  N,  et  L,  ductaque  perpendiculari  secante 
unam  curvam  in  X,  et  alteram  in  S,  aequales  forent  (ex 
nota  hujus  curvae  natura)  ipsae  QX,  QS,  quod  est  ab- 
surdum.  Idem  valebit,  si  in  dicta  superpositione  maneat 
in  suo  situ  recta  NF,  et  recta  AC  cadat  super  LF.    Rur- 


ai6 


corresponding  parts,  of  which  it  is  certain  they  are  equal 
to  each  other. 

Now  for  the  proof  of  the  proposition. 

Suppose  erected  f  rom  the  base  AB  to  the  curve  CKD 
(fig.  45)  indefinitely  many  perpendiculars  NF,  LF,  PF, 
MK,  TF,  VF,  IF;  and  on  the  base  AB  take  as  equal  the 
portions  AN,  NL,  LP,  PM,  MT,  TV,  VI,  IB. 


R- 

< 

f- 

r 

r 

^ 

f\ 

1 

Fig.  45. 

First  is  certain  the  angle  of  AC  with  the  curve  will 
be  equal  to  each  of  the  angles  of  the  aforesaid  perpen- 
diculars  with  the  curve  on  either  side  at  the  points  F,  or 
at  the  point  K,  or  at  the  point  D.  For  if  the  mixed 
quadrilateral  [91]  ANFC  is  supposed  to  be  superposed 
upon  the  mixed  quadrilateral  NLFF,  the  base  AN  lying 
upon  the  equal  base  NL,  AC  falls  upon  NF,  and  NF 
upon  LF,  because  of  the  equal  right  angles  at  the  points 
A,  N,  L.  Then  because  of  the  equaHty  (from  P 
XXXIV.)  of  the  straights  AC,  NF,  LF,  the  point  C  falls 
upon  point  F  of  NF,  and  this  upon  the  other  point  F 
of  LF. 

Moreover  the  curve  CF  exactly  fits  the  curve  FF: 
for  if  one  of  these,  as  CF  fell  within  or  without;  any 
point  Q  being  assumed  between  points  N,  and  L,  and  the 
perpendicular  being  drawn  cutting  one  curve  in  X,  and 
the  other  in  S,  QX,  QS  would  be  equal  ( f  rom  the  known 
nature  of  this  curve),  which  is  absurd.  The  same  will 
hold,  if  in  the  said  superposition  the  straight  NF  remains 
in  its  place,  and  the  straight  AC  falls  upon  LF.     Again 


117 


sum  idem  valebit,  si  idem  quadrilaterum  mistum  ANFC 
utrovis  modo  superponi  intelligatur  cuivis  reliquorum 
quadrilaterorum  usque  ad  ipsum  inclusive  postremum 
quadrilaterum  BDFL  Itaque  angulus  ipsius  AC  cuni  ea 
curva  aequalis  est  singulis  hinc  inde  ad  puncta  F,  sive  ad 
punctum  K,  aut  punctum  D,  praedictarum  perpendicula- 
rium  angulis  cum  eadem  curva. 

Constat  hinc  secundo  aequales  adamussim  inter  se  esse 
portiones  ipsius  curvae  ab  istis  perpendicularibus  hinc 
inde  abscissas. 

Si  ergo  basis  AB  divisa  sit  bifariam  in  M,  et  medie- 
tas  AM  bifariam  in  L;  tum  quadrans  LM  bifariam  in  P; 
atque  ita  in  infinitum,  procedendo  semper  versus  partes 
puncti  M;  constabit  tertio,  etiam  curvam  CKD  bifariam 
dividi  in  K  a  perpendiculari  MK,  medietatem  CK  bifa- 
riam  itidem  dividi  in  F  a  perpendiculari  LF,  quadrantem 
FK  bifariam  in  F  a  perpendiculari  PF;  atque  ita  in  infi- 
nitum,  procedendo  semper  uniformiter  versus  partes  ip- 
sius  puncti  K. 

Quoniam  vero  in  ista  basis  AB  in  infinitum  divisione 
[92]  considerare  possumus  rem  devenisse  ad  portionem 
ipsius  AB  infinite  parvam,  quae  nempe  exhibeatur  per 
latitudinem  infinite  parvam  perpendicularis  MK,  constabit 
quarto  (ex  praemisso  axiomate)  aequalitas  intenta  totius 
basis  AB  cum  tota  curva  CKD,  dum  alias  ostendam  por- 
tionem  infinite  parvam  abscissam  ex  basi  AB  a  perpen- 
diculari  MK  aequalem  esse  adamussim  portioni  infinite 
parvae,  quam  eadem  perpendicularis  abscindit  ex  curva 
CKD.     Et  hoc  quidem  postremum  sic  demonstro. 

Nam  RK  perpendicularis  ipsi  KM  tanget  (ex  35. 
hujus)  curvam  in  K,  atque  ita  eandem  tanget  in  K,  ut  in- 
ter  ipsam  tangentem  (ex  Cor.  post  Z6.  hujus)  et  curvam, 
ex  neutra  parte  locari  possit  recta,  quae  ipsam  curvam  non 


218 


the  same  will  hold,  if  the  sanie  mixed  quadrilateral  ANFC 
in  either  mode  is  supposed  to  be  superposed  to  any  of  the 
remaining  quadrilaterals  even  to  the  last  quadrilateral 
BDFI  inclusive. 

Therefore  the  angle  of  AC  with  the  curve  is  equal  to 
either  of  the  angles  with  this  curve  of  the  aforesaid 
perpendiculars  on  either  side  at  the  points  F,  or  at  the 
point  K,  or  point  D. 

Hence  follows  secondly  that  the  portions  of  the  curve 
cut  off  on  each  side  by  these  perpendiculars  are  exactly 
equal  to  one  another. 

If  therefore  the  base  AB  be  bisected  in  M,  and  the 
half  AM  bisected  in  L;  then  the  quarter  LM  bisected  hi 
P;  and  so  in  infinitum,  proceeding  always  toward  the 
parts  of  the  point  M ;  it  will  follow  thirdly,  also  the  curve 
CKD  is  bisected  in  K  by  the  perpendicular  MK,  the  half 
CK  in  hke  manner  bisected  in  F  by  the  perpendicular  LF, 
the  quarter  FK  bisected  in  F  by  the  perpendicular  PF ; 
and  so  in  infinitwn,  proceeding  always  uniformly  toward 
the  parts  of  the  point  K. 

But  since  in  this  division  of  the  base  AB  in  infinitum 
we  may  [92]  consider  the  thing  to  have  arrived  at  a  portion 
of  AB  infinitely  small,  which  obviously  may  be  exhibited 
by  the  infinitely  small  breadth  of  the  perpendicular  MK, 
fourthly  (from  the  premised  axiom)  will  follow  the  as- 
serted  equahty  of  the  whole  base  AB  with  the  whole  curve 
CKD.  if  only  I  now  can  show  the  infinitely  small  portion 
cut  off  from  the  base  AB  by  the  perpendicular  MK  to  be 
exactly  equal  to  the  infinitely  small  portion,  which  the 
same  perpendicular  cuts  off  from  the  curve  CKD. 

And  this  last  I  thus  demonstrate. 

For  RK  perpendicular  to  KM  touches  (from  P. 
XXXV.)  the  curve  at  K,  and  touches  this  in  K  so,  that 
between  the  tangent  (from  Cor.  to  P.  XXXVI.)  and  the 
curve  from  neither  side  can  be  placed  a  straight,  which 

219 


secet.  Igitur  infinitesima  K,  spectans  ad  curvam,  aequalis 
omnino  erit  infinitesimae  K  spectanti  ad  tangentem.  Con- 
stat  autem  infinitesimam  K  spectantem  ad  tangentem,  nec 
majorem,  nec  minorem,  sed  omnino  aequalem  esse  infini- 
tesimae  M  spectanti  ad  basim  AB;  quia  nempe  recta  illa 
MK  intelligi  potest  descripta  ex  fluxu  semper  ex  aequo 
ejusdem  puncti  M  usque  ad  eam  summitatem  K. 

Quare  (juxta  praemissum  axioma)  curva  CKD,  ex 
hypothesi  anguH  acuti  enascens,  aequahs  esse  deberet  con- 
trapositae  basi  AB.     Quod  erat  demonstrandum. 

SCHOLION  I. 

Sed  forte  minus  evidens  cuipiam  videbitur  enunciata 
exactissima  aequalitas  inter  illas  infinitesimas  M,  et  K. 
Quare  ad  avertendum  hunc  scrupulum  sic  rursum  pro- 
cedo.  Cuidam  rectae  AB  insistant  ad  rectos  angulos  in 
eodem  plano  (fig.  48.)  duae  rectae  aequales  AC,  BD. 
Rursum  in  eodem  plano  intehigatur  existere  circulus 
BLDH,  cujus  diameter  BD;  sitque  semicircumferentia 
BLD  aequa-[93]hs  praedictae  AB.  Praeterea  idem  cir- 
culus  ita  in  eo  plano  revolvi  concipiatur  super  ea  recta 
AB,  ut  motu  semper  continuo,  et  aequabih  perficiat,  seu 
describat  suae  ipsius  semicircumferentiae  punctis  prae- 
dictam  B A ;    quousque  nempe  punctum  D,  ad  illam  semi- 


does  not  cut  the  curve.  Therefore  infinitesimal  K,  re- 
garding  the  curve,  will  be  wholly  equal  to  infinitesimal  K 
regarding  the  tangent.  But  it  is  certain  the  infinitesimal 
K  regarding  the  tangent  is  neither  greater  nor  less  than, 
but  exactly  equal  to  the  infinitesimal  M  regarding  the 
base  AB;  because  obviously  the  straight  MK  may  be 
supposed  described  by  the  flow  always  uniform  of  the 
point  M  up  to  the  summit  K. 

Wherefore  (according  to  the  premised  axiom)  the 
curve  CKD,  born  of  the  hypothesis  of  acute  angle  should 
be  equal  to  the  opposite  base  AB. 

Quod  erat  demonstrandum. 


SCHOLION  I. 

But  perchance  to  some  one  will  seem  by  no  means 
evident  the  enunciated  exact  equahty  between  the  infini- 
tesimals  M,  and  K.  Wherefore  to  remove  this  scruple 
I  again  proceed  thus. 

To  a  certain  straight  AB  let  two  equal  straights  AC, 
BD  (fig.  48)  stand  at  right  angles  in  the  same  plane. 
c 


Again  in  the  same  plane  suppose  there  is  a  circle 
BLDH,  whose  diameter  is  BD;  and  let  the  semicircum- 
ference  BLD  be  equal  [93]  to  the  aforesaid  AB.  Further 
let  the  same  circle  be  conceived  so  to  be  revolved  in  that 
plane  upon  the  straight  AB  that  with  motion  always 
continuous  and  uniform  it  achieves  or  describes  with  the 
points  of  its  semicircumference  the  aforesaid  BA,  until 


circumferentiam  spectans,  perveniat  ad  congruendum  ipsi 
puncto  A,  ita  ut  propterea  punctum  B,  ejusdem  semicir- 
cumferentiae  alterum  extremum  punctum,  deveniat  ad 
congruendum  illi  puncto  C. 

His  stantibus ;  si  in  semicircumf erentia  BLD  designe- 
tur  quodvis  punctum  L,  cui  in  descripta  recta  linea  BA 
correspondeat  punctum  M,  ex  quo  in  eo  tali  plano  edu- 
catur  perpendicularis  MK,  aequalis  ipsi  BD :  Dico  illud 
punctum  K  fore  ipsum  punctum  H  diametraliter  opposi- 
tum  illi  puncto  L.  Nam  ibi  in  puncto  M,  sive  L  recta  AB 
continget  praedictum  circulum.  Igitur  MK  eidem  AB 
perpendicularis  transibit  (ex  19.  tertii,  quae  utique  in- 
dependens  est  ab  Axiomate  controverso)  per  centrum 
ejusdem  circuli.  Quare ;  ubi  punctum  L  in  ea  tali  circuli 
BLDH  revolutione  perveniat  ad  congruendum  cum 
puncto  M  ipsius  AB,  etiam  punctum  H,  diametraliter 
oppositum  praedicto  puncto  L,  incidet  in  punctum  K 
illius  MK. 

Porro  constat  idem  similiter  valere  de  reliquis  punctis 
semicircumferentiae  BLD,  et  horum  diametraliter  cor- 
relativis  in  altera  semicircumferentia  BHD.  Quare  linea, 
eo  tali  modo  successive  descripta  a  punctis  semicircum- 
ferentiae  BHD,  erit  illa  eadem  jam  expensa  DKC,  quae 
nempe  suis  omnibus  punctis  aequidistet  ab  illa  recta  BA; 
sitque  idcirco  (juxta  hypothesin  anguli  acuti)  semper 
cava  versus  partes  ejusdem  AB. 

Inde  autem  fit,  ut  punctum  M  in  ea  BA  censendum  sit 
exactissime  aequale  puncto  K  in  altera  DKC,  propter[94] 
omnimodam  istorum  aequalitatem  cum  punctis  L,  et  H 
diametraliter  oppositis  in  eo  circulo  BLDH.  Quare ;  cum 
idem  valeat  de  omnibus  punctis  descriptae  rectae  BA,  si 
conferantur  cum  aliis  uniformiter  contrapositis  in  prae- 
dicta  supposita  curva  DKC ;  consequens  plane  est,  ut  ipsa 


indeed  point  D  pertaining  to  that  semicircumference 
comes  to  congriience  with  point  A,  so  that  moreover 
point  B,  the  other  extreme  point  of  the  same  semicircum- 
ference  comes  to  congruence  with  point  C. 

This  abiding;  if  in  the  semicircumference  BLD  is 
designated  any  point  L,  to  which  in  the  described  straight 
line  BA  corresponds  point  M,  from  which  in  that  plane 
is  erected  the  perpendicular  MK,  equal  to  BD :  I  say  that 
point  K  will  be  the  point  H  diametrically  opposite  the 
point  L. 

For  there  in  the  point  M,  or  L  the  straight  AB 
touches  the  aforesaid  circle.  Therefore  MK  perpendicu- 
lar  to  AB  will  go  (from  Eu.  IIL  19,  which  is  assuredly 
independent  of  the  controverted  axiom)  through  the 
center  of  the  same  circle.  Wheref ore ;  where  point  L  in 
that  revolution  of  the  circle  BLDH  comes  to  congruence 
with  the  point  M  of  AB,  also  point  H,  diametrically  oppo- 
site  the  aforesaid  point  L,  falls  upon  point  K  of  MK. 

Furthermore  it  is  certain  the  same  holds  in  Hke  manner 
of  the  remaining  points  of  the  semicircumference  BLD, 
and  of  those  diametrically  correlative  in  the  other  semicir- 
cumference  BHD.  Wherefore  the  line,  in  that  way  succes- 
sively  described  by  the  points  of  the  semicircumference 
BHD,  will  be  the  ah-eady  considered  DKC,  which  in  all 
its  points  is  equidistant  f rom  the  straight  B A ;  and  which 
therefore  (in  accordance  with  the  hypothesis  of  acute 
angle)  is  always  concave  toward  the  side  of  AB. 

But  thence  follows,  that  the  point  M  in  BA  may  be 
considered  exactly  equal  to  point  K  in  DKC,  because  of 
[94]  their  equality  in  every  way  with  the  points  L,  and  H 
diametrically  opposite  in  the  circle  BLDH. 

Wherefore ;  since  the  same  holds  of  all  points  of  the 
described  straight  BA,  if  they  be  compared  with  the  other 
uniformly  opposite  in  the  aforesaid  assumed  curve  DKC; 
the  consequence  evidently  is,  that  this  curve,  born  of  the 


223 


talis  curva,  ex  hypothesi  anguH  acuti  enascens,  censenda 
sit  aequaHs  contrapositae  basi  AB.  Atque  id  est,  quod 
nova  hac  methodo  iterum  demonstrandum  susceperam. 


SCHOLION  II. 

Rursum"  vero :  quoniam  recta  BA  intehigitur  succes- 
sive  descripta  a  punctis  semicircumferentiae  BLD  motu 
iHo  semper  aequabiH,  et  continuo;  cui  nempe  descriptioni 
correspondet  descriptio  ihius  Hneae  DKC  a  punctis  dia- 
metraliter  correlativis  alterius  semicircumferentiae  BHD : 
obvium  est  intehigere,  quod  ipsa  recta  BA  motu  iUo  sem- 
per  aequabiH,  et  continuo  describatur  ab  eo  unico  puncto 
B,  quod  nempe  (veluti  repHcatum)  inteUigatur  cum  ipsa 
taH  semicircumferentia  semper  excurrere  super  ea  BA; 
dum  interim  eodem  ipso  tempore,  motu  eodem  semper 
aequabiH,  et  continuo,  describitur  iha  altera  DKC  ab 
altero  diametraHter  correlativo  unico  puncto  D,  quod 
ipsum  rursum  (vektti  repHcatum)  inteHigatur  cum  sua 
altera  semicircumferentia  BHD  semper  excurrere  super 
praedicta  DKC.  Tunc  autem  faciHus  intehigitur  intenta 
aequaHtas  inter  eam  DKC,  et  eidem  contrapositam  rectam 
BA;  quippe  quae  duae  aequaH  ipso  tempore,  et  aequaH 
motu  intehiguntur  descriptae  a  duobus  exactissime  inter 
se  aequaHbus  punctis,  seu  mavis  infinitesimis.  Ubi  constat 
hanc  ipsam  exactissimam  praedictorum  punctorum  aequa- 
Htatem  non  esse  mihi  in  ista  nova  contemplatione  neces- 
sariam.  [95J 

PROPOSITIO  XXXVIII. 

Hypothesis  anguli  acuti  est  ahsolute  falsa,  quia  se  ipsam 
destruit. 

Demonstratur.  Nam  supra  ex  ipsa  hypothesi  anguH 
acuti  evidenter  eHcuimus,  curvam  CKD  (fig.  46.)  ex  ea 


hypothesis  of  acute  angle,  is  to  be  thought  equal  to  the 
opposite  base  AB. 

And  that  is  what  I  had  undertaken  again  to  demon- 
strate  by  this  new  method. 

SCHOLION  II. 

But  again :  since  the  straight  B A  is  discerned  as  suc- 
cessively  described  by  the  points  of  the  semicircumference 
BLD  by  that  motion  always  uniform  and  continuous ;  to 
which  description  corresponds  the  description  of  that  Hne 
DKC  by  the  diametrically  correlative  points  of  the  other 
semicircumference  BHD:  it  is  easy  to  understand,  that 
this  straight  BA  by  that  motion  always  uniform,  and  con- 
tinuous  is  described  by  the  one  point  B,  which  of  course 
(as  if  unrolled)  is  thought  always  to  run  out  with  that 
semicircumf erence  upon  BA ;  whilst  meanwhile  in  exactly 
the  same  time,  by  the  same  motion  always  uniform,  and 
continuous,  is  described  that  other  DKC  by  the  other  one 
diametrically  correlative  point  D,  which  again  itself  (as 
if  unrolled)  is  thought  with  its  other  semicircumference 
BHD  always  to  run  out  upon  the  aforesaid  DKC. 

But  then  is  more  easily  understood  the  asserted  equal- 
ity  between  DKC,  and  the  straight  BA  opposite  it;  since 
the  two  are  imagined  to  be  described  in  equal  time,  and 
equal  motion  by  two  exactly  equal  points,  or,  if  you  pre- 
fer,  infinitesimals. 

Where  it  holds  that  this  exact  equality  of  the  af  oresaid 
points  is  not  necessary  for  me  in  that  new  considera- 
tion.  [95] 

PROPOSITION  XXXVIII. 

The  hypothesis  of  acufe  angle  is  absolutely  false,  hecause 
it  destroys  itself. 

Proof.  Assuredly  we  have  above  clearly  deduced 
from  the  hypothesis  of  acute  angle,  that  the  curve  CKD 


prognatam  aequalem  esse  debere  contrapositae  basi  AB. 
Nunc  autem  contradictorium  ex  eadem  hypothesi  elici- 
mus,  quod  curva  CKD  nequeat  esse  aequalis  illi  basi,  cum 
certe  sit  eadem  major.  Quod  enim  curva  CKD  major  sit 
recta  CD  ejus  extremitates  jungente,  notio  est  omnibus 
communis,  quam  etiam  demonstrare  possumus  ex  vige- 
sima  primi,  quod  duo  trianguli  latera  reliquo  semper  sunt 
majora;  junctis  nimirum  CK,  et  KD;  ac  rursum  junctis 
similiter  apicibus,  primo  quidem  duorum,  tum  quatuor, 
et  sic  in  infinitum,  duplicato  numero  enascentium  seg- 
mentorum,  quousque  intelHgatur  hoc  pacto  absumi,  seu 
desinere  in  ipsas  infinite  parvas  seu  chordas,  seu  tan- 
gentes,  tota  curva  CKD.  Sed  hic  procedere  possumus  ex 
sola  communi  notione.  Quod  autem  juncta  CD  major 
sit  basi  AB,-  demonstratum  a  nobis  est  in  3.  hujus  ex  ipsis 
visceribus  hypothesis  anguH  acuti.  Igitur  curva  CKD, 
ex  hypothesi  anguH  acuti  enascens,  est  certe  major  basi 
AB,  quia  est  major,  saltem  ^x  communi  notione,  recta 
CD,  quae  ex  hac  ipsa  hypothesi  anguH  acuti  demonstratur 
major  basi  AB.  Non  igitur  potest  simul  consistere,  quod 
curva  ista  CKD  aequaHs  sit  basi  AB.  Itaque  constat 
hypothesim  anguH  acuti  esse  absolute  falsam,  quia  se 
ipsam  destruit. 


«a6 


Fig.  46. 


(fig.  46)  born  of  it  must  be  equal  to  the  opposite  base 
AB.  But  now  we  deduce  the  contradictory  from  the 
same  hypothesis,  that  the  curve 
CKD  cannot  be  equal  to  that 
base,  since  surely  it  is  greater 
than  it. 

For  that  the  curve  CKD  is 
greater  than  the  straight  CD 
joining  its  extremities,  the  no- 
tion  is  common  to  all,  which  also 
we  may  demonstrate  from  Eu. 
I.  20,  that  two  sides  of  a  triangle 
are  always  greater  than  the  third;  join  CK,  and  KD; 
and  again  join  Hkewise  the  apices,  first  of  two,  then  of 
four,  and  so  on  in  infinitum,  the  number  of  the  produced 
segments  doubling,  until  the  whole  curve  CKD  is  under- 
stood  in  this  way  to  be  exhausted,  or  to  end  in  those 
infinitely  small  chords,  or  tangents. 

However  here  we  may  proceed  from  the  common 
notion  alone. 

But  that  the  join  CD  is  greater  than  the  base  AB,  has 
been  demonstrated  by  us  in  P.  III.  f  rom  the  very  viscera 
of  the  hypothesis  of  acute  angle. 

Therefore  the  curve  CKD,  born  of  the  hypothesis  of 
acute  angle,  is  certainly  greater  than  the  base  AB,  because 
it  is  greater,  anyhow  from  the  common  notion,  than  the 
straight  CD,  which  from  the  hypothesis  of  acute  angle 
is  demonstrated  greater  than  the  base  AB.  Therefore 
cannot  at  the  same  time  stand,  that  the  curve  CKD  is 
equal  to  the  base  AB. 

Consequently  is  established  that  the  hypothesis  of 
acute  angle  is  absolutely  false,  because  it  destroys  itself. 


a»7 


SCHOLION. 
Observare  tamen  debeo,  quod  etiam  ex  hypothesi  an- 
guli  obtusi  enascitur  curva  quaedam  CKD,  sed  con-[96] 
vexa  versus  partes  basis  AB.  Nam  MH  (fig.  47.)  bi- 
fariam  dividens  ipsas  AB,  CD  erit  (ex  2.  hujus)  eisdem 
perpendicularis ;  et  major  (ex  Cor.  I.  post  3.  hujus)  ipsis 
AC,  BD,  in  hypothesi  anguH  obtusi.  Quare  ipsius  MH 
portio  quaedam  MK  aequahs  erit  ipsi  AC,  aut  BD.  Tum 
junctis  CK,  et  KD,  divisisque  bifariam  in  punctis  X,  P, 
Q,  N  rectis  CK,  AM,  MB,  KD,  constat  (ex  eadem  2. 
hujus)  junctas  PX,  QN,  perpendiculares  fore  ipsis  rectis 
divisis.  At  rursum  erunt  illae  (ex  eodem  Cor.  I.  post  3. 
hujus)  majores  ipsis  AC,  MK,  BD.  Hinc;  assumptis 
earundem  portionibus  PL,  QS,  quae  praedictis  aequales 
sint ;  habebimus  curvam,  ex  hypothesi  anguli  obtusi  enas- 
centem,  quae  transibit  per  puncta  C,  L,  K,  S,  D.  Atque 
ita  semper,  si  decernere  veHmus  reliqua  puncta  ejusdem 
curvae.  Inde  autem  constat  eam  fore  convexam  versus 
partes  basis  AB.  Jam  fateor  demonstrari  uniformi  plane 
methodo  potuisse  aequahtatem  hujus  curvae  cum  ipsa 
basi  AB.  At  quis  f  ructus  ?  Nullus  sane.  Quemadmodum 
enim  curva  ista  CKD  censeri  debet,  ex  communi  saltem 
notione,  major  recta  CD;  ita  etiam  (in  3.  hujus)  basis 
AB  demonstratur  major  eadem  CD,  dum  stet  hypothesis 
anguli  obtusi.  Nullum  ergo  ex  hac  parte  absurdum,  si 
basis  AB  aequaHs  sit  praedictae  curvae.    AHter  vero  rem 


328 


H 


n 


Fig.  47. 


SCHOLION. 

I  should  still  observe,  that  also  from  the  hypothesis 
of  obtuse  angle  is  born  a  certain  curve  CKD,  but  convex 
[96]  toward  the  side  of  the  base 
AB. 

For  MH  (fig.  47)  bisecting 
AB,  CD  will  be  (from  P.  II.) 
perpendicular  to  them;  and 
greater  (from  Cor.  I.  to  P.  III.) 
than  AC,  BD,  in  the  hypothesis 
of  obtuse  angle. 

Wherefore  a  certain  portion  MK  of  MH  will  be  equal 
to  AC,  or  BD. 

Then  CK  and  KD  being  joined,  and  the  straights 
CK,  AM,  MB,  KD  bisected  in  the  points  X,  P,  Q,  N, 
it  follows  (again  from  P.  II.)  that  the  joins  PX,  QN 
will  be  perpendicular  to  the  divided  straights. 

But  again  they  will  be  (from  the  same  Cor.  I.  to  P. 
III.)  greater  than  AC,  MK,  BD. 

Hence;  taking  of  them  the  portions  PL,  QS,  which 
are  equal  to  the  aforesaid;  we  shall  have  a  curve,  bom 
of  the  hypothesis  of  obtuse  angle,  which  will  go  through 
the  points  C,  L,  K,  S,  D.  And  so  on  always,  if  we  wish 
to  determine  remaining  points  of  the  same  curve. 

But  thence  follows  it  will  be  convex  toward  the  side 
of  the  base  AB.  Now  I  grant  in  just  the  same  way 
could  have  been  demonstrated  the  equality  of  this  curve 
with  its  base  AB.    But  what  good  ?    None  at  all. 

For  just  as  the  curve  CKD  must  be  thought,  anyhow^ 
from  the  common  notion,  greater  than  the  straight  CD; 
so  also  (in  P.  III.)  the  base  AB  is  proved  greater  than 
CD,  when  the  hypothesis  of  obtuse  angle  holds.  There- 
fore  from  this  side  is  nothing  absurd,  if  the  base  AB  be 
cqual  to  the  aforesaid  curve. 


procedere  in  hypothesi  anguU  acuti,  constat  ex  dictis 
supra. 

Ex  hoc  igitur  SchoHo,  et  ex  altero  post  13.  hujus  in- 
telHgi  potest,  diversam  plane  viam  iniri  debuisse  ad  refel- 
lendam  utranque  f  alsam  hypothesim,  unam  anguli  obtusi, 
et  alteram  anguH  acuti. 

Praeterea  facile  itidem  est  ex  istis  dignoscere,  non 
nisi  rectam  Hneam  CD  esse  posse,  quae  omnibus  suis 
punctis  aequidistet  ab  ea  supposita  recta  Hnea  AB.  [97] 

PROPOSITIO  XXXIX. 

Si  in  duas  rectas  lineas  altera  recta  incidens,  internos  ad 
easdemque  partes  angulos  duobus  rectis  minores  fa- 
ciat,  duae  illae  rectae  lineae  in  infinitum  productae 
sibi  mutuo  incident  ad  eas  partes,  ubi  sunt  anguli 
duobus  rectis  niinores. 

Et  hoc  est  notum  iHud  Axioma  EucHdaeum,  quod 
tandem  absolute  demonstrandum  suscipio.  Ad  hunc 
autem  finem  satis  erit  recolere  nonnuUas  praecedentium 
Demonstrationum.  Itaque  in  meis  Propositionibus,  usque 
ad  VII.  hujus  inclusive,  tres  secrevi  hypotheses  circa  rec- 
tam  jungentem  extrema  puncta  duorum  aequaUum  per- 
pendiculorum,  quae  uni  cuidam  rectae,  quam  basim  ap- 
peUo,  in  eodem  plano  insistant.  Porro  circa  has  hypo- 
theses  (quas  invicem  secerno  ex  specie  angulorum,  qui  ad 
eam  jungentem  fieri  censeantur)  demonstro  unam  quam- 
Hbet  earum,  nimirum  aut  anguH  recti,  aut  anguH  obtusi, 
aut  anguH  acuti,  si  vel  in  uno  casu  sit  vera,  semper  et  in 
omni  casu  iUam  solam  esse  veram.  Tum  in  XIII.  ostendo 
universalem  veritatem  Axiomatis  controversi,  dum  con- 
sistat  alterutra  hypothesis  aut  anguH  recti,  aut  anguH  ob- 


230 


But  that  the  thing  goes  otherwise  in  the  hypothesis  of 
acute  angle,  follows  from  what  is  said  above. 

From  this  schoHon  therefore  and  f rom  the  other  after 
P.  XIII.  may  be  reaHzed,  that  a  whoUy  different  way 
was  to  be  taken  in  refuting  each  false  hypothesis,  one  of 
obtuse  angle,  and  the  other  of  acute  angle. 

Moreover  it  is  easy  in  Hke  manner  to  recognize  f  rom 
these,  that  it  can  only  be  a  straight  Hne  CD,  which  in  all 
its  points  is  equidistant  from  the  assumed  straight  Hne 
AB.  m 

PROPOSITION  XXXIX. 

//  upon  two  straight  lines  another  straight  striking  makes 
toward  the  same  parts  angles  less  than  two  right 
angles,  those  two  straight  lines  produced  in  infinitum 
meet  each  other  toward  those  parts  where  are  the 
angles  less  than  two  right  angles. 

This  is  the  famous  EucHdean  axiom,  which  at  length 
I  undertake  absolutely  to  demonstrate. 

For  this  end  however  it  wiU  be  sufficient  to  recall 
some  of  the  preceding  demonstrations.  Therefore  in  my 
propositions,  up  to  P.  VII.  inclusive,  I  have  distinguished 
three  hypotheses  about  the  straight  joining  the  extreme 
points  of  two  equal  perpendiculars,  which  stand  upon 
a  certain  straight,  that  I  caH  base,  in  the  same  plane. 

Furthermore  in  regard  to  these  hypotheses  (which  in 
turn  I  distinguish  from  the  species  of  the  angles,  which 
are  supposed  to  be  made  at  the  join)  I  demonstrate  that 
any  one  of  them,  forsooth  either  of  right  angle,  or  obtuse 
angle,  or  acute  angle.  alone  is  true  always  and  in  every 
case,  if  even  in  one  case  it  be  true. 

Then  in  P.  XIII.  I  show  the  universal  truth  of  the 
controverted  axiom,  when  occurs  either  the  hypothesis 
of  right  angle,  or  of  obtuse  angle. 

2Zl 


tusi.  Hinc  in  XIV,  declaro  absolutam  falsitatem  hypo- 
thesis  anguli  obtusi,  quia  se  ipsam  destruentis,  utpote  quae 
praedicti  Axiomatis  veritatem  infert,  ex  quo  contra  reli- 
quas  duas  hypotheses  soH  hypothesi  anguH  recti  locus 
rehnquitur.  Igitur  sola  restat  hypothesis  anguH  acuti, 
contra  quam  diutius  pugnandum  f uit. 

Et  hujus  quidem  (post  multa,  ne  dicam  omnia,  con- 
ditionate  expensa)  absolutam  falsitatem  in  XXXIII.  tan- 
dem  ostendo,  quia  repugnantis  naturae  Hneae  rectae,  circa 
quam  multa  ibi  intersero  necessaria  Lemmata.  Tandem 
vero  in  praecedente  Propositione  absolute  demonstro  sibi 
ipsi  repugnantem  hypothesin  anguH  acuti.  Quoniam  igi- 
[98]tur  unica  restat  hypothesis  anguH  recti,  consequens 
plane  est,  ut  ex  praedicta  XIII.  hujus  stabiHtum  absolute 
maneat  praenunciatum  EucHdaeum  Axioma.  Quod  erat 
propositum. 

SCHOLION. 

Sed  juvat  expendere  hoc  loco  notabile  discrimen  inter 
praemissas  duarum  hypothesium  redargutiones.  Nam 
circa  hypothesin  anguH  obtusi  res  est  meridiana  luce  cla- 
rior ;  quandoquidem  ex  ea  assumpta  ut  vera  demonstratur 
absoluta  universaHs  veritas  controversi  Pronunciati  EucH- 
daei,  ex  quo  postea  demonstratur  absoluta  falsitas  ipsius 
taHs  hypothesis;  prout  constat  ex  XIII.  et  XIV.  hujus. 
Contra  vero  non  devenio  ad  probandam  f  alsitatem  alterius 
hypothesis,  quae  est  anguH  acuti,  nisi  prius  demonstrando ; 
quod  Hnea,  cujus  omnia  puncta  aequidistent  a  quadam 
supposita  recta  Hnea  in  eodem  cum  ipsa  plano  existente, 
aequaHs  sit  ipsi  taH  rectae ;  quod  ipsum  tamen  non  videor 


»3* 


Hence  in  P.  XIV.  I  declare  the  absolute  falsity  of  the 
hypothesis  of  obtuse  angle,  because  it  destroys  itself, 
inasmuch  as  it  occasions  the  truth  of  the  aforesaid  axiom, 
from  which  against  the  remaining  two  hypotheses  place 
is  left  for  the  hypothesis  of  right  angle  alone.  Therefore 
remains  only  the  hypothesis  of  acute  angle,  against  which 
was  longer  to  be  fought. 

And  of  this  indeed  (after  many  things,  I  do  not  say 
all,  circumstantially  considered)  at  length  in  P.  XXXIII. 
I  show  the  absolute  f alsity,  because  repugnant  to  the  na- 
ture  of  the  straight  line,  about  which  I  there  introduce 
many  necessary  lemmata. 

Finally  in  the  preceding  proposition  I  absolutely  prove 
the  hypothesis  of  acute  angle  contradictory  to  itself. 

Since  therefore  [98]  the  hypothesis  of  right  angle 
alone  remains,  the  consequence  plainly  is,  that  from  the 
aforesaid  P.  XIII.  remains  absolutely  established  the 
enunciated  EucHdean  axiom. 

Quod  erat  propositum. 

SCHOLION. 

It  is  well  to  consider  here  a  notable  difference  between 
the  foregoing  refutations  of  the  two  hypotheses.  For 
in  regard  to  the  hypothesis  of  obtuse  angle  the  thing 
is  clearer  than  midday  light;  since  from  it  assumed  as 
true  is  demonstrated  the  absolute  universal  truth  of  the 
controverted  Euclidean  postulate,  from  which  afterward 
is  demonstrated  the  absolute  falsity  of  this  hypothesis; 
as  is  established  from  P.  XIII.  and  P.  XIV. 

But  on  the  contrary  I  do  not  attain  to  proving  the 
falsity  of  the  other  hypothesis,  that  of  acute  angle,  with- 
out  previously  proving;  that  the  line,  all  of  whose  points 
are  equidistant  from  an  assumed  straight  line  lying  in 
the  same  plane  with  it,  is  equal  to  this  straight,  which 
itself  finally  I  do  not  appear  to  demonstrate  from  the 


demonstrare  ex  visceribus  ipsiusmet  hypothesis,  prout 
opus  foret  ad  perfectam  redargutionem. 

Respondeo  autem  tripHci  medio  usum  me  fuisse  in 
XXXVII.  hujus  ad  demonstrandam  praedictam  aequali- 
tatem.  Et  primo  quidem,  in  corpore  ilHus  Propositionis, 
demonstro  eam  curvam  CKD,  prout  enascentem  ex  hypo- 
thesi  anguH  acuti  (ac  propterea  semper  cavam  versus 
partes  iHius  rectae  AB)  aequalem  eidem  esse  debere,  et 
quidem  argumentum  ducendo  ex  ipsis  ejusdem  curvae 
tangentibus.  Deinde  in  duobus  ejusdem  Propositionis 
subsequentibus  SchoHis,  praecisive  a  quaHbet  speciaH  hypo- 
thesi,  bis  rursum  demonstro  aequaHtatem  iUius  genitae 
Hneae  CD  cum  subjecta  recta  Hnea  AB,  quaHscunque  tan- 
dem  censeatur  esse  ipsa  Hnea  CD  eo  modo  genita. 

Jam  vero;  quatenus  iHa  curva  CKD,  prout  enascens 
[99]  ex  hypothesi  anguH  acuti,  censeatur  primo  iUo  modo 
demonstrata  aequaHs  subjectae  rectae  Hneae  AB;  mani- 
festa  evadit  redargutio,  cum  ex  eadem  hypothesi  eviden- 
ter  demonstretur  major.  Sin  autem  alterutro  ex  duobus 
aHis  modis  ostensa  censeatur  aequaHtas  praedicta;  neque 
tunc  cessat  redargutio  contra  hypothesin  anguH  acuti. 
Ratio  est ;  quia  nihil  vetat,  quin  iHa  CD  sit  curva,  et  nihi- 
lominus  aequaHs  sit  iUi  rectae  AB,  dum  tamen  sit  semper 
versus  eas  partes  convexa,  ac  propterea  recta  jungens  illa 
eadem  puncta  C,  et  D  minor  sit  contraposita  basi  AB, 
prout  in  hypothesi  anguH  obtusi :  At  omnino  repugnat,  si 
versus  easdem  partes  sit  semper  cava,  ac  propterea  recta 
jungens  praedicta  iHa  puncta  C,  et  D  major  sit  eadem 
contraposita  basi  AB,  prout  in  hypothesi  anguH  acuti. 
Atque  ita  declaratum  jam  est  in  SchoHo  praecedentis 
Propositionis.     Scilicet  contra  hypothesin  anguH  obtusi 


»34 


viscera  of  the  very  hypothesis,  as  must  be  done  for  a 
perfect  refutation. 

But  I  reply  I  used  a  triple  means  in  P.  XXXVII.  for 
demonstrating  the  mentioned  equaHty. 

And  first,  in  the  body  of  the  proposition,  I  prove  the 
curve  CKD,  as  born  from  the  hypothesis  of  acute  angle 
(and  therefore  always  concave  toward  the  side  of  the 
straight  AB)  must  be  equal  to  it,  and  indeed  by  drawing 
the  argument  from  the  tangents  of  the  curve. 

Then  in  two  subsequent  schoHa  of  the  proposition, 
apart  from  any  special  hypothesis,  twice  again  I  demon- 
strate  the  equaHty  of  the  generated  Hne  CD  with  the 
underlying  straight  Hne  AB,  of  whatever  kind  the  Hne 
CD  so  generated  is  supposed  to  be. 

But  now;  in  so  far  as  the  curve  CKD,  as  born  [99] 
f  rom  the  hypothesis  of  acute  angle,  is  judged  to  be  proved 
by  the  first  method  equal  to  the  underlying  straight  Hne 
AB,  a  manifest  refutation  arises,  since  from  the  same 
hypothesis  it  is  evidently  proved  greater.  But  if  the 
aforesaid  equaHty  is  supposed  shown  in  either  of  the  two 
other  modes;  not  even  then  does  the  refutation  cease 
against  the  hypothesis  of  acute  angle.  The  reason  is; 
because  nothing  forbids,  that  CD  may  be  curved,  and 
nevertheless  may  be  equal  to  the  straight  AB,  while  yet 
it  may  be  always  convex  toward  that  side,  and  therefore 
the  straight  joining  the  points  C,  and  D  may  be  less 
than  the  opposite  base  AB,  as  in  the  hypothesis  of  obtuse 
angle.  But,  it  is  whoHy  contradictory,  if  toward  that 
side  it  be  always  concave,  and  therefore  the  straight 
joining  the  points  C,  and  D  be  greater  than  the  opposite 
base  AB,  as  in  the  hypothesis  of  acute  angle. 

And  so  has  just  now  been  stated  in  the  schoHon  of 
the  preceding  proposition. 

Of  course  against  the  hypothesis  of  obtuse  angle  it 


235 


manifestum  est  nullam  hinc  sequi  redargutionem,  quae 
propterea  unice  impetit  hypothesin  anguli  acuti. 

Hoc  tamen  loco  aliquis  fortasse  inquiret,  cur  adeo 
solHcitus  sim  in  demonstranda  utriusque  f alsae  hypothesis 
exacta  redargutione.  Ad  eum,  inquam,  finem,  ut  inde 
magis  constet  non  sine  causa  assumptum  f  uisse  ab  Euclide 
tanquam  per  se  notum  celebre  illud  Axioma.  Nam  hic 
maxime  videtur  esse  cujusque  primae  veritatis  veluti  cha- 
racter,  ut  non  nisi  exquisita  aliqua  redargutione,  ex  suo 
ipsius  contradictorio,  assumpto  ut  vero,  illa  ipsa  sibi  tan- 
dem  restitui  possit.  Atque  ita  a  prima  usque  aetate  mihi 
feliciter  contigisse  circa  examen  primarum  quarundam 
veritatum  profiteri  possum,  prout  constat  ex  mea  Logica 
demonstrativa. 

Inde  autem  transire  possum  ad  explicandum,  cur  in 
Proemio  ad  Lectorem  dixerim:  non  sine  magno  in  rigv- 
dam  Logicam.  peccato  assumptas  a  quibmdam  fuisse  tan- 
quam  datas  [100]  duas  rectas  lineas  aequidistantes.  Ubi 
monere  debeo  nullum  eorum  a  me  hic  carpi,  quos  in  hoc 
meo  Libro  vel  indirecte  nominavi,  quia  vere  magnos  Geo- 
metras,  hujusque  peccati  verissime  immunes.  Dico  autem : 
magnum  in  rigidam  Logicam  peccatum :  quid  enim  aliud 
est  assumere  tanquam  datas  duas  rectas  lineas  aequidistanr 
tes :  nisi  aut  velle ;  quod  omnis  Hnea  in  eodem  plano  aequi- 
distans  a  quadam  supposita  Hnea  recta  sit  ipsa  etiam  Hnea 
recta;  aut  saltem  supponere,  quod  una  aHqua  sic  aequi- 
distans  possit  esse  Hnea  recta,  quam  idcirco  seu  per  hypo- 
thesin,  seu  per  postulatum  praesumere  Hceat  in  tanta 
aHqua  unius  ab  altera  distantia?  At  constat  neutrum 
horum  venditari  posse  tanquam  per  se  notum.  SciHcet 
ratio  objectiva  Hneae,  quae  omnibus  suis  punctis  aequi- 
distet  a  quadam  supposita  Hnea  recta,  non  ita  clare  per 


»96 


is  manifest  no  refutation  follows  hence,  which  therefore 
only  demolishes  the  hypothesis  of  acute  angle. 

In  this  place  however  some  one  perchance  may  inquire, 
why  I  am  so  soHcitous  about  proving  exact  the  refutation 
of  each  false  hypothesis.  To  the  end,  say  I,  that  thence 
may  be  more  completely  established  that  not  without 
cause  was  that  famous  axiom  assumed  by  EucHd  as 
known  per  se.  For  chiefly  this  seems  to  be  as  it  were 
the  character  of  every  primal  verity,  that  precisely  by  a 
certain  recondite  argumentation  based  upon  its  very  con- 
tradictory,  assumed  as  true,  it  can  be  at  length  brought 
back  to  its  own  self.  And  I  can  avow  that  thus  it  has 
turned  out  happily  for  me  right  on  f rom  early  youth  in 
reference  to  the  consideration  of  certain  primal  verities, 
as  is  known  from  my  Logica  demonstrativa. 

Thence  now  I  may  proceed  to  explain,  why  in  the 
Preface  to  the  Reader  I  have  said:  not  without  a  great 
sin  against  rigid  logic  two  equidistant  straight  lines  have 
been  assumed  by  some  as  given.  [100] 

Where  I  should  point  out  that  none  of  those  is  carped 
at,  whom  I  have  mentioned  even  indirectly  in  this  book 
of  mine,  because  they  are  truly  great  geometers,  and 
verily  f ree  from  this  sin. 

But  I  say :  great  sin  against  rigid  logic :  f  or  what  else 
is  it  to  assume  as  given  two  equidistant  straight  lines: 
unless  either  to  assume ;  that  every  Hne  equidistant  in  the 
same  plane  from  a  certain  supposed  straight  Hne  is  itself 
also  a  straight  line ;  or  at  least  to  suppose,  that  some  one 
thus  equidistant  may  be  a  straight  line,  as  if  therefore  it 
were  allowable  to  make  assumption,  whether  by  hypoth- 
esis,  or  by  postulate,  of  any  such  distance  of  one  from 
another?  But  it  is  certain  neither  of  these  can  be  made 
traffic  of  as  if  per  se  known. 

Forsooth  the  objective  concept  of  a  line,  which  in  all 
its  points  is  equidistant  from  a  certain  supposed  straight 

»37 


se  ipsam  congruit  cum  definitione  propria  ipsius  lineae 
rectae.  Quare  assumere  duas  rectas  lineas  sub  ista  aequi- 
distantiae  ratione  inter  se  parallelas  fallacia  est,  quam  in 
praedicta  mea  Logica  appello  Definitionis  complexae, 
juxta  quam  irritus  est  omnis  progressus  ad  assequendam 
veritatem  absolute  talem. 

Unam  tamen  superesse  adhuc  video  necessariam  ob- 
servationem.  Nam  lineam  jungentem  extrema  puncta 
omnium  aequalium  perpendiculorum,  quae  in  eodem  plano 
versus  easdem  partes  erigantur  a  singulis  punctis  sub- 
jectae  rectae  lineae  AB,  debere  esse  et  aequalem  prae- 
dictae  AB,  et  rursum  in  seipsa  rectam,  fateri  omnes  volu- 
mus.  Sed  dico  prius  esse  apud  nos,  quod  aequalis  sit; 
deinde  autem,  quod  recta.  Cum  enim  singula  puncta  illius 
rectae  AB  intelligi  possint  semper  aequabiliter  procedere 
per  sua  illa  perpendicula  ad  formandam  tandem  illam 
qualemcunque  CD;  manifestum  videri  debet,  quod  illa 
qualiscunque  genita  CD  aequalis  sit  eidem  AB;  praeser- 
tim  vero,  si  respiciamus  explicationem  contentam  in 
Scholio  11.  post  [101]  XXXVII.  hujus,  ubi  hoc  punctum 
clarissime  demonstratum  est. 

Sed  postea  magna  adhuc  restat  difficultas  in  demon- 
strando,  quod  illa  eadem  sic  genita  CD  non  nisi  recta 
linea  sit.  Atque  hinc  factum  esse  puto,  ut  ex  communi 
quadam  persuasione  rectam  lineam,  pro  faciliore  pro- 
gressu,  maluerint  praesumere,  ut  inde  aequalem  osten- 
derent  illi  basi  AB,  ac  postea  inferrent  rectos  angulos  ad 
ipsam  talem  jungentem  CD.  Dico  autem  magnam  diffi- 
cultatem :  Nam  prius  expendere  oportebat  tres  hypotheses 
circa  angulos  ad  illam  junctam  rectam  CD,  nimirum  aut 
rectos,  si  ipsa  aequalis  sit  basi  AB ;  aut  obtusos,  si  minor ; 


238 


line,  clearly  is  not  thus  per  se  congruent  with  the  proper 
definition  of  the  straight  Hne. 

Wherefore  to  define  two  parallel  straight  lines  under 
this  relation  of  mutual  equidistance  is  the  fallacy,  which 
in  my  aforesaid  Logica  I  call  definitionis  complexae,  in 
connection  with  which  every  advance  toward  attaining 
truth  absolutely  such  is  ineffectual. 

I  see  in  addition  there  still  remains  one  necessary  ob- 
servation. 

For  we  all  are  wilHng  to  grant  the  Hne  joining  the 
extreme  points  of  ah  equal  perpendiculars,  which  in  the 
same  plane  are  erected  toward  the  same  parts  f  rom  the 
separate  points  of  an  underlying  straight  Hne  AB,  must 
be  both  equal  to  the  aforesaid  AB,  and  moreover  in  itself 
straight. 

But  I  say  with  us  is  first,  that  it  is  equal;  then  how- 
ever,  that  it  is  straight. 

For  since  the  single  points  of  the  straight  AB  may  be 
thought  always  to  proceed  uniformly  upon  those  perpen- 
diculars  of  theirs  to  forming  at  length  that  certain  CD; 
it  should  seem  manifest,  that  the  generated  CD,  of  what- 
soever  kind,  is  equal  to  AB ;  but  especiahy,  if  we  consider 
the  expHcation  contained  in  SchoHon  II.  after  [101]  P. 
XXXVII. ,  where  this  point  is  most  clearly  demonstrated. 

But  thereafter  still  remains  a  great  difficulty  in  demon- 
strating,  that  this  same  generated  CD  cannot  be  anything 
but  a  straight  line.  And  hence  comes  it  I  think,  that  f  rom 
a  certain  common  conviction,  for  more  facile  progress, 
they  have  preferred  to  presume  the  line  straight,  that 
thence  they  might  show  it  equal  to  the  base  AB,  and 
afterward  infer  right  angles  at  the  join  CD. 

But  I  say  great  difficidty:  For  first  it  was  necessary 
to  consider  three  hypotheses  about  the  angles  at  the 
straight  join  CD,  forsooth  either  right,  if  it  be  equal  to 
the  base  AB;  or  obtuse,  if  less;  or  acute,  if  greater.    But 

239 


aut  acutos,  si  major.  Tum  vero  ostendi  debebat  non  nisi 
cavam  esse  posse  versus  basim  AB  lineam  curvam,  quae 
(in  hypothesi  anguli  acuti)  jungat  extremitates  illorum 
aequalium  perpendiculorum,  ac  rursum  non  nisi  con- 
vexam  versus  eandem  basim  aham  curvam,  quae  (in 
hypothesi  anguli  obtusi)  jungat  extremitates  eorundem 
perpendiculorum.  Deinde  autem  hypothesis  quidem  an- 
guli  acuti  ex  eo  demonstranda  erat  falsa;  quia  hnea  jun- 
gens  praedictorum  perpendiculorum  extremitates  adeo 
non  erit  aequahs  basi  AB,  ut  immo  (ex  communi  saltem 
notione)  major  sit  illa  juncta  recta  CD,  quae  ex  natura 
ipsiusmet  hypothesis  major  est  praedicta  basi  AB.  At 
hypothesis  anguH  obtusi  aHunde  ostendenda  erat  sibi  ipsi 
repugnans,  prout  in  XIV.  hujus.    Sed  haec  jam  satis. 

Finis  Libri  primi. 


then  it  had  to  be  shown  that  the  curved  line,  which  (in 
the  hypothesis  of  acute  angle)  joins  the  extremities  of 
those  equal  perpendiculars,  could  only  be  concave  toward 
the  base  AB,  and  again  the  other  curve,  which  (in  the 
hypothesis  of  obtuse  angle)  joins  the  extremities  of  the 
same  perpendiculars,  only  convex  toward  the  same  base. 
But  then  the  hypothesis  indeed  of  acute  angle  from  this 
was  demonstrated  false ;  because  the  Hne  joining  the  ex- 
tremities  of  the  aforesaid  perpendiculars  was  so  far  not 
equal  to  the  base  AB,  as  on  the  contrary  (anyhow  from 
the  common  notion)  it  is  greater  than  the  straight  join 
CD,  which  from  the  nature  of  this  hypothesis  itself  is 
greater  than  the  aforesaid  base  AB. 

But  the  hypothesis  of  obtuse  angle  had  to  be  shown 
from  another  source  contradictory  to  itself,  as  in  P.  XIV. 

But  this  now  is  enough. 

End  of  Book  I. 


«41 


NOTES. 

Page  21.  Prop.  IIT:  Euclid's  first  two  postulates  are: 
Let  it  be  granted, 

1.  that  one  and  only  one  sect  can  be  drawn  from  any 
point  to  any  other ; 

2.  and  that  this  sect  may  be  produced  continually  on  its 
straight. 

In  I.  16  he  assumes  that  the  straight  divides  the  plane 
into  two  separate  regions,  and  also  the  Archimedes  assump- 
tion  that  the  straight  is  infinite  and  open.  This  block  of 
assumptions  is  incompatible  with  the  hypothesis  of  obtuse 
angle  as  Saccheri  later  shows.  If  it  were  also  incompatible 
with  the  hypothesis  of  acute  angle,  we  should  have  a  perfect 
case  of  Saccheri's  favorite  method.  The  proofs  would  be 
fairy  proofs  leading  to  a  direct  demonstration  of  their  con- 
tradictory  opposite ;  and  none  of  them  could  make  part  of  a 
modern  treatise  on  non-Euchdean  geometry. 

But  since  Euclid's  assumptions,  barring  the  Parallel  Pos- 
tulate,  are  perfectly  compatible  with  the  hypothesis  of  acute 
angle,  many  of  Saccheri's  proofs  remain  the  most  elegant 
and  cogent  the  world  possesses  in  the  domain  of  non- 
Euclidean  geometry. 

Page  27.  Prop.  III,  Cor.  II :  Saccheri  simply  cites  this 
corollary  when,  as  often,  he  wishes  the  proposition:  In  any 
birectangular  quadrilateral  HMPC  with  angle  P  obtuse  and 
angle  C  acute,  side  PM  <  CH. 

To  this  the  proof  of  Prop.  III,  Part  3,  applies. 

243 


Page  33.  Prop.  VI:  Here  is  assumed  the  principle  of 
continuity.  An  elementary  proof  without  this  and  without 
the  Archimedes  assumption  is  given  by  Bonola. 

Without  these,  and  with  only  a  sect-carrier  replacing  the 
circle  in  constructions,  Euclid's  geometry  and  a  geometry 
f  ulfilHng  the  hypothesis  of  obtuse  angle  are  given  in  Halsted, 
Geometrie  rationnelle,  Paris,  Gauthier-Villars.  Compare,  for 
the  hypothesis  of  acute  angle:  John  Bolyai,  The  Science 
Absolute  of  Space,  translated  from  the  Latin  by  Dr.  George 
Bruce  Halsted ;  and  Nicholas  Lobachevski,  Geometrical  Re- 
searches  on  the  Theory  of  Parallels,  translated  from  the 
original  by  George  Bruce  Halsted  (The  Open  Court  Pub- 
lishing  Company,  1914). 

Page  95 f.  Demonstrations  physico-geometric.  If  in  a 
single  case  the  angle  inscribed  in  a  semicircle  be  ascertained 
to  be  right,  EucHdean  geometry  is  estabHshed.  But  measure- 
ment  being  imperfect,  this  is  hopeless.  What  if  such  angle 
were  f ound  other  than  right  by  a  difference  greater  than  the 
Hmits  of  experimental  error? 

Consult:  George  Bruce  Halsted,  The  Foundations  of 
Science,  New  York,  The  Science  Press,  1913. 

Page  109.  Prop.  XXI.  SchoHon  IV:  Saccheri  misses 
the  possibiHty  that  the  intersection  point  P  of  APY  and 
XPY  may  go  to  infinity  while  AX  remains  finite. 

Page  192.  Ly  is  a  term  of  the  grammarians  and  rhetori- 
cians,  by  which  is  denoted  the  treatment  of  a  word  as  itself 
a  thing.    The  Greek  article  to  was  thus  used. 

Page  221.  Prop.  XXXVII.  SchoHon  I:  Is  it  possible 
Saccheri  did  not  perceive  that  his  reasoning  appHes  just  as 
weH  to  proving  two  concentric  circles  equal? 

Page  224.  Prop.  XXXVII.  Scholion  II:  "aequaH  tem- 
pore,"  yes ;  but  "aequaH  motu"  is  unproven. 


SUBJECT  INDEX. 


Angle  in  a  semicircle,   ix  f,   75,   244. 

Angle-sum,  viii,  61,   67. 

Assumption,    Archimedes's,    xi,    xxix, 

243  f. 
Assumption,    Clavius's,   ix,    91. 
Assumptions,  Euclid's,   xi,  xxviii  f,  243. 
Asymptotic  straights,   ix. 

Birectangular    quadrilateral,    viii,    243. 

Chrysippaean   Syllogism,   xxii. 
Compatibility,  xix  fF,  243. 
Conchoid,  83. 
Continuity,    244. 


Inscribed  angle,  x,  75. 
Italian  translation,  viii. 

Logica    demonstrativa,    vii,    xv,    xviii, 

xxiv. 
Ly,    192,   244 

Monist,   viii. 

Parallel   Postulate,   xxiv  f,   xxviii  f,    5, 

231. 
Physico-geometric    demonstrations,    13, 

95,   244. 
Plane,    179. 
Postulate  V,  xi. 


Definition  of  parallels,  ix,   7. 
Definitions,  xviii  f. 


Right  angle,  201. 
Right-angled  triangle,   41 


Equidistant,  ix,  209. 
Euclid  IX.    12,  xxiii  f. 
Euclid  III.  31,  ix. 
Exterior  angle,   39. 


Saccheri'3  Theorem,   viii. 
Similarity,    105. 
Spherical  geometry,  xi. 
Square,  xviii. 
Straight  line,   ix,    173. 


Incompatibility,    xix  ff,    243. 
Independence,   xxi. 


Triangle,    angle-sum,   61,    67. 
Triangle,    exterior   angle,    39. 


345 


INDEX  OF  PROPER  NAMES. 


Apollonius  (co,  250-200),  5. 
Archimedes     (287-212),     xi,    xxix,     5, 

243  f  • 
Aristotle    (384-322),   xvii. 
Augustae  Taurinorum,  see  Turin. 
Augustse  Ubiorum,  see  Cologne. 

Baker,  Alfred,  x. 
Barbarin,   Paul,   vii. 
Beltrami,  E.  (1835-1900),  viii. 
Boccardini,   G.,  viii. 
Bolyai,  John   (1802- 1860),  xi,  244. 
Eonola,  R,   (1875-1911),  viii,  244. 
Borelli,   Giovanni   A.    (1608- 1679),   ix, 
XX,   87. 

Cardan   (1501-1576),  xxi  f,  xxiv. 
Casalette,    Francesco,    Count    Gravere, 

XV  f . 
Cayley,    Arthur    (1821-1895),    xii. 
Ceva,    Giovanni    (1648- 1736),   viii,   xv. 
Ceva,  Tommaso,  xv,  xxiv. 
Clavius,    C.    (i 537-1612),    ix,    xi,    xxii, 

xxiv,   5,  83,  85,  87,  91,  99,  201. 
Clifford,    William    K.    (1845-1879),    x. 
Cologne,  vii,   xvii. 
Coolidge,  Julian  L.,  viii. 

Dante  (1265-1321),  x, 

Euclid    (ca.  330-275),  passim. 

Geminos  (ca.   100-40),  ix. 

Genoa,  xv. 

Gravere,    Count,    see   Casalette. 

Heath,    Sir  Thomas  L.,  vii,  xi,  xviii. 
Humphreys,  Milton  W.,  xii  f. 
Huntington,  Edward  V.,  xxi. 


Kelland,   Philip,  xi. 


Lobachevski,    Nicholas    (1793-1856), 
244. 

Manganotti,  A.,  vii  f. 
Mansion,   P.    (1844-1919),  vii  f . 
Milan,  xv  f. 

Mill,   J.    S.    (1806-1873),   xviiif. 
Moore,   Robert  L.,  xxi. 

Nasiraddin  (1201-1274),  loi,  103,  137. 
Nicomedes  of  Gerasa   (ca.   180  A.  D.), 
83. 

Oxford,  xxviii,    105. 

Pascal,   Alberto,  vii  f,   xv. 
Pavia,  XV  f,  xxi. 
Perry,  John,  xii. 
Proclus    (412-485),    83. 

San  Remo,  xv. 

Savile,   Sir  Henry   (1549-1622),  xxviii. 

Segre,  Corrado,  xxx. 

Simson,  Robert,  ix,  xi. 

Stanhope,    P.,   vii. 

Stone,    Edmund,    x. 

Thales    (640-548),   201. 
Theodosius  of   Tripoli,   xxii,    5. 
Ticinum  Regium,  see  Pavia. 
Turin,  xv  f. 

Vailati,    Giovanni    (1863-1909),    xv, 

xix  f,  xxv  f. 
Veblen,   Oswald,  xxi. 
Victor    Amadeus   II    (1666-1732),    xv. 
Vitale,  Giordano  (1633-1711),  ix,  xxix. 

Wallace,   Alfred  R.    (1822-1912),  x. 
Wallis,  John   (1616-1703),   loi,   103, 

105. 
Whithers,  John  W.,   xii. 


246 


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